Engine stalls / hesitations at idle and low acceleration

Model: 1999 Hyundai Elantra GLS 2.0l DOHC
Odometer: 78,000
Recent work: replaced MAF with new, cleaned ignition plate, ran through 1 tank with Techron fuel system cleaner.

The problem has a few variations:
- most often: you're cruising at 25-30, and the engine is doing little work; then, the engine goes quiet, and the gas pedal stops producing power -- the car starts slowing down, and soon after the engine shuts off
- less often: power cuts out as you're accelerating after being stopped at a signal, or as you're going through a turn at low speed -- these are dangerous situations, because you cannot maneuver until the engine restarts
- a couple of times: you've stopped at a signal, and the engine briefly runs rough or even stops

Lately, the engine has been sounding smooth through all this, with the only symptoms being the sudden loss of gas pedal response, the engine going quiet, and engine shutoff soon after that.

Sometimes, but not always, one can recover from the problem by quickly pressing down the gas. On a successful recovery, this is followed by a half-second of silence, followed by a rev of the engine and a partial recovery of power. However, after powering through such hiccups, the car is very prone to having more problems in the next few minutes. Sometimes, such as in tight turns, it is too dangerous to try to recover by rapid acceleration, and in these cases, the engine goes dead if not revved soon after the hesitation.

In a recent instance, the car was repeatedly losing power over about 10 minutes:

1) While gradually accelerating into a left turn on a 4-lane road, after starting from a signal. I was driving for about 3 minutes after having the car parked in a sunny spot on a warm day for 2 hours. I pulled over into the left lane, turned off the engine, and restarted the car. Sometimes, the problem goes away after such a treatment, but in this case, it did not.

2) A minute or two later, I was accelerating from a signal, and midway through the intersection, the car completely stopped responding to the gas pedal (I pumped it a few times). I was trying to change into the right lane with the remaining 5-10mph of speed, right after the light. Right-lane traffic was not letting me go, so I was basically stopped. And then at one last desperate pump of the gas pedal, the engine revved and recovered.

3) For the remainder of the drive, I tried to rev the engine to above 2500-3000rpm (hard to say without an indicator) regularly, and the car did not stall, although it did seem a bit iffy.

In another recent instance, I was about 20 minutes into looking for parking in a crowded area (so lots of slow driving), in warm sunny weather, when the engine stalled suddenly as I was making a slow right turn into a slight incline. It restarted with no subsequent issues.

In another recent instance, the car was parked in 55 F weather, at night (no rain), for about an hour after a 20-minute drive, and about 2 minutes into the return drive, it went through a series of hesitations and near-stalls (saved by accelerating), until finally coming to a dead stop on a stop sign. There were no further issues after re-starting the engine.

On occasion, the problem happens after a cold start in the morning, but more frequently (as described above) it happens after driving somewhere, and leaving the car for a couple of hours. It has happened both in chilly weather and hot.

Despite all this, most of the time, the car runs smoothly, and several shops have declined to look at it because they don't see any issues on casual examination. They say: If I cannot see it, I cannot fix it. However, it does tend to happen at least every 30 miles, and sometimes much more frequently (practically every day for the past few days).

The first time the problem happened was several years ago, but then it did not recur for 2-3 years. It started in earnest, and has been gradually getting worse over the last year. No check engine light or warning light is on, and the mechanics see no codes on the computer.

There problem _might_ be more likely to occur on a nearly-empty tank. At any rate, in the last 1.5 tanks of gas, all the major problems were happening when the tank was under 1/4 full, with smaller hesitations and almost no stalls in the top 25% of the tank. I am not sure if this effect is for real, or just a recent coincidence.