Put the key in the ignition and turned the key, no starter noise?

Ken,

Severe tendonitis has kept me off my feet for the past 10 days, so that accounts for this delayed response.

Your questions--
1. Instrument panel VATS warning does not flash and disappears normally after about 1-2 seconds. This behavior seems completely normal, despite the fact the bypass resistor's current value (on VOM test, and more than one year after the bypass was installed) now shows only 1331 Ohms, instead of the bypass resistor's original 2,000 Ohm value. The ignition key Hall-Effect resistor, from which the proper bypass resistor value was estimated, still shows 1861 Ohms. Since the bypass circuit does not produce a sustained VATS warning, should I replace the 1331 Ohm resistor, anyway?
2. My idea about replacing the starter enabler replay was simple deduction, since the starter, itself, cranks, but the starter enable (trigger) produces no current when the ignition key is turned to "start".
3. Thank you for additional information. What still puzzles me is which electrical connector you mean-- the specific location is still elusive. I expected to find a starter relay module in clear view, near the steering column. The diagram specifically positions the relay to the left of the steering column support. Tomorrow, if I can, I'll try another series of photos, to show you where my search is focused.

Ken,

Thanks for the word on the resistor. Having spent many hours diagnosing a VATS security lock-out, it did not make sense that if the resistor was faulty, the VATS warning was not active.
Regarding power through the ignition switch, I checked for end-of-circuit power through the ignition switch using an assistant to press the brake pedal while in Park and turn the key to start. Absolutely no voltage showed on my VOM, connected directly to the S connector (as described above in an earlier post).
As you point out, the problem lies somewhere in the circuit input for ignition. That is why I tried the "obvious" option of checking the starter enable relay. That relay is still invisible, despite many evenings of twisting myself onto the under-dash floorboard to search more closely.
The most specific information I have was found just hours ago, and I have spent as much time as I had taking photos and searching, according to the information. That information says, "The (starter-enabled) relay is found under the left-hand of the instrument panel, above the instrument panel fuse block. Look just left of the steering column support and on a bracket beside the main wiring conduit. You must remove the fuse block cover and panel."
* I have labeled two of the attached photos for your convenience, but hope they are not confusing in their orientation.

If I can confirm the relay is "above" the main electrical connector-- that is, if I can find it-- replacement of the relay would seem in reach, at last. In the next 24 hours, I must plunder through my collection of auto tools to find the circuit tester.
At this point, I should mention I have located something of interest near the main electrical connector. With my fingers and a small lantern flashlight, I spotted a small cylindrical part, about the size of the relay cube. But its cylindrical/pot shape does not match either the four- or five-bladed relay cubes I bought from O'Reilly (making sure to specify Buick98, 3.8L, LeSabre Custom).
Just a thought-- do you have the ability to use Windows Paint (in every version of Windows) to draw an arrow in the photo to the part you believe may be the relay? Editing an arrow into a photo may be unfamiliar territory for you but could be a practical way for you to show me where the relay hides.

* BTW, I have confirmed I can right-click and "Save As" on any of the photos I posted, and download the photo to my desktop for editing. So, I presume you can do the same to download the photo file, edit an arrow into the photo, and then re-post the edited photo in your response.

** As I mentioned earlier, my tendonitis healed slowly, and I still cannot walk very far, so I understand the value of having a spare car or two. I appreciate your help immensely.

Here is a photo of the starter enable relay, but of two models, one with four blades, the other with five (and a steel retainer clip).
Note the small inset text box, with relay brand and each model number.

Here is another photo of the starter enable relay, which comes in two models-- one with four blades, and the other with five blades (and a steel retainer clip on the relay body).
Note the small inset text box, with relay brand and each model number, which I tried to make more legible.

Thanks, I'll look behind the main electrical connector, next. What threw me off was the source I quoted, saying the relay is on a bracket beside the main connector-- not behind it.

Ken,

Despite lingering tendonitis, my feet and ankles are improved enough, I went back to work on the car this morning. Wedging into the left-side, below-dash floorboard area. I located the main electrical connector, with a plan to remove its single hex screw (7mm), but the screw-- even loosened to max-- did not allow the nylon connector to be removed. Nonetheless, I used my fingers to reach behind the main electrical connector but found nothing there.
Reviewing my work, I have looked everywhere on the right and left sides of the steering column, and am familiar with the location of the instrument panel fuse block and of the main electrical connector. Nowhere have I found anything resembling the two starter enable relays (4- and 5-blade) I purchased.
Quite a few forums address various GM issues, and some offer schematics which may vary from GM model to model. Somehow, my Buick 98 starter enable relay location is clearly visible on the schematic you sent (in a previous post), yet cannot be located visually in any of the forum offerings. The location you arrowed for me in a photo as a likely position turned out to have nothing behind it.

What I need, at this point, is a clear photo, not a line drawing, which helps orient me to the actual location. Photos submitted by owners/users are usually not very helpful, and there are very few sharp and clearly defined enough to contribute anything.
Am I giving up? Of course, not. I am stuck with this problem. The only way out is through.

The fuse and relay panel layout does not match my vehicle-- but it was a good, intuitive try. Would the VIN help? VIN 1G4HP52K5WH500185
I have attached a photo of the below dash, inside MiniFuse panel (driver left), showing no relay cubes. Tomorrow, or whenever I can catch a break in the rain, I'll take another, better photo of the inside fuse panel (115A) and the outside MaxiFuse panel and outside relay panel layouts.
BTW-- Months ago, before all this happened, I purchased a PDF copy of the actual shop manual, but even that does not have specifics on the relay location.
We know one thing-- the Buick 98 LeSabre 3.8L has a starter enable relay, and its location is clearly indicated in the line drawing and schematic you have provided and which I have found elsewhere, too. But no photos, of course.
You indicated earlier it is time to trace the ignition circuit. What do you think about this approach?-- From the schematic, identify the two lines leading to and from the starter enable relay, then trace each line physically from each end of the circuit back to the relay. Somehow, the incoming pair and outgoing pair ought to coincide at the relay location. I know from the schematic there is a yellow (Y) and purple (PPL) wire pair entering and exiting the relay. That indicates the enable relay is (probably) a four-blade unit, although the line drawing clearly shows a metal clip attached to the relay body and suggests a five-blade relay unit.
Since I never have traced auto circuits before, I need to spend some time outside looking at the schematic, then tracing the lines.
A persistent weather low is parked over the SE US where I live, and I may be in for some continuous rain delay. All to say, I appreciate you're not giving up on me as I struggle through this.

Updating you: My tendonitis has vanished, but the rain continues. Despite that, I have traced the positive line from the battery to the relay and Maxifuse bank, and this is where the fun begins-- at the power distribution bus.

According to all references I have read, the starter enable relay is inside, beneath the dash, but just to be sure, I first looked over each relay or fuse on the firewall power panel, and found no label referred to a starter.

Looking beneath the dash, again, I had no better luck, so far. So, the rest is tracing down from the steering column, to reach the circuit contact points for the starter trigger circuit.

In that regard, a question-- Is it necessary to pull the steering column housing? That is, do you know of a better way to check contact points from the ignition switch?

Thanks, Ken, I'll look for it.
Of course, it is perfectly logical that no accurate Buick-issued instructions for the relay location exist, but another, unlabeled Buick view might contain exactly what we need. Hope so, anyway.
Should I remove anything on the steering column and/or its support bracket to reach this location? I know it's largely up to me to determine that, but I just wondered.

Ken,

Is there any possibility the VATS bypass that I created, using a 2k Ohm resistor, could be faulty, since its latest Ohm value (by VOM test) is down to 1334 Ohms? The difference between the resistor's installed Ohm value and the much lower current value is my only reason for asking.
Of course, it would not make sense for my VATS bypass resistor to be the problem-- even if of the wrong value. Currently, the VATS system seems normal-- its "Security" warning does not flash, but stays solid for about five seconds, and then disappears. Besides, I operated the car with no cranking problem when the resistor was below the ignition key's 1861 Ohm value.

Another update--
1. I checked thoroughly with flashlight and mirror but saw nothing which matched the size and shape of the starter enable relay purchased from the manager of my local O'Reilly auto parts store. However, while searching in exactly the area you and others suggested, I did find an oddly shaped plastic connector which had four wires running from it-- two light yellow, one light yellow with black stripe and one green. Since the connector was firmly in-place, I decided not to apply major force until I understood which way to pull or pry.
2. Now, the interesting part. I had put away all my tools and was on the way inside the house, when I stopped, recalling the image of something odd I had seen beneath the dash, in the Mini fuse panel. I turned around, unlocked the car, and looked carefully at Mini fuse location 2E, which I had presumed was empty on first inspection (not remembering this is the starter enable circuit fuse location). Looking more closely, I saw-- inside each leg of the socket-- was what appeared to be the silver stump of a Mini fuse.
Apparently, fuse 2E had blown/melted, and its red plastic cap had fallen off, leaving only the two metal remnants of the fuse, deep in the socket. Tomorrow, I'll use a turkey basting needle to probe inside the socket to pry out the fuse leg stump. If I am successful with that operation, the next step is to find another 10A fuse, or borrow one from the spares in the fuse panel and see what happens.