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

Yes, I can send you a photo of the Mini fuse panel, showing the glint of what remains of the fuse, inside the socket.
The newly discovered problem is a melted fusible link, but not melted wiring.
The loss of fuse 2E could explain everything-- provided the engine resumes normal operations.
Of course, I would like to find the plastic red cap showing "10A", but the important thing is to remove what I presume to be the two fuse link 'legs" from their respective sockets.

Here is the promised photo.
View of damaged Mini Fuse 2E, showing two metallic stumps remaining in their socket. Apparently, the major part of the fuse body, including red plastic cap, simply fell away from its socket.

The plan is to use a sharp, needle-like probe to pry out each of the two stumps. The problem with that approach is gaining full access to the Mini Fuse socket panel, which faces downward, toward the floor.

1. The Mini Fuse panel is not an array of separate sockets, so I cannot remove only socket 2E.

2. Disconnection of the airbag and all Mini Fuse wires to release the Mini Fuse panel connections seems a very awkward solution.

The lower left dash shroud contains a mount for the Mini Fuse panel, but fingertip exploration has not revealed how the Mini Fuse panel attaches to the mount. The Mini Fuse panel does not seem to slide or snap into place on the mount.

You mean the MiniFuses's two blades do not appear to be on the same line? All I am able to see is what the camera shows. Which illustrates the problem-- I need to have better access to socket 2E. To make the operation simpler, I would like to detach all wires to the lower dash shroud that do not connect to the Minifuse panel, reorient the newly freed shroud and work on the Minifuse panel from a better angle. However, working access is not the only problem.

At the moment, I must work with the lower dash shroud in my left hand, turning its corner down to expose the Minifuse panel, I cannot see very much of the area of interest. At first, I attempted to use a surgical clamp to pull out what appears to be a stump or remnant of the original Minifuse blade but could not. After that, I tried to push a straight pick through the back of the socket, hoping to engage the metal sump, but so far, there is not enough room for the pick to slide past the wire in the socket.

It seems the only way to prove Minifuse 2E is the key to restarting the Buick is to remove the two wires entering the socket from the back-- a large diameter white wire, and a small diameter dark-brown wire-- and attach these securely to a 10A red-capped Minifuse. I probably could solder them onto the Minifuse short-term, or even longer-term (failing to remove the two metal stumps).

Is there a special puller for already inserted Minifuse panel wires?

Ken,
You will be pleased to know the problem has been resolved.
1. Apparently, the 10A red-capped MiniFuse was not properly in place in the original non-crank event, for some undetermined reason. The most likely reason for finding the fuse out of socket was it did fall out, because it had been improperly inserted. Yet, I still find that hard to understand-- I know how a properly inserted fuse should feel as I push it into place. No explanation, yet.
2. I tested with an identical fuse from the airbag location 2C. Insertion of the fuse into the MiniFuse panel was easy-- no mechanical resistance, which means the original socket contacts are still okay.
Sorry to burden you with the long explanation, but you deserve a great deal of credit for hanging with this developing situation.

Many thanks!

Ken,
More evidence appeared this afternoon on the cause of the non-crank problem. This afternoon, when I restarted the engine after buying new MiniFuses at a local store, I remembered I had borrowed the 10A fuse from the airbag slot, and an airbag might turn out to be important, later. So, I turned off the engine, and inserted a 10A fuse into the MiniFuse airbag slot, but carefully and slowly. As I did so, the fuse seemed so loose, it might eventually fall out.
This could have happened with slot 2E, as well. As proof, when I tried to restart the engine, it failed to crank, just as with the original non-crank event. I removed and carefully examined the 2E fuse for any faults, but found none, except for the fact this Mini"Fuse is made of a very thin metal conductor. I reversed the orientation of the fuse, and made sure it was fully inserted and secure. The reversal of fuse position permitted the engine to restart.
Apparently, the MiniFuse panel sockets on this Buick 98 no longer "grip" the fuses are they once did, and produce weird and difficult diagnostics unless the malfunction is on the list of possible contributing conditions.

Ken, is the procedure to tighten the terminal tension described anywhere?

Thanks for the follow up, Ken.
The MiniFuse dropout surprised me most-- I did not know such things could happen.
Surely, hundreds of Buick/Olds and other GM drivers of the MiniFuse persuasion are subject to the same, very hidden problem.
Your video will be interesting, though sufficiently informative shots (with a full view of what your fingers are doing) may be a challenge, by the very nature of this operation.

Ken, Last week the same MiniFuse location 2E caused a non-crank situation. I removed the fuse, reinserted it, and the engine started with no problem.
Although I am on notice about this grumpy, electrically-challlenged old car, surely others face the same situation with MiniFuses. Your forthcoming video should get substantial attention.

Ken,
Thanks for all your help during the past year !
Merry Christmas and Happy New 2024!
BG