1987 Toyota Celica stumbles with acceleration

[Malibu67]

Hi-have a 1987 Toyota Celica, 2.0L 3SFE, automatic transmission, 108171 miles.

Last month, my car would start up and run fine. However, since last month, after 5-10 minutes of residential driving (30 MPH) the car would hesitate when accelerating from a stop. It would hesitate for a few seconds and then pick up power again. Took my car to a mechanic. They had it for 4 days- couldn't get the car to present the problem. Diagnostic and Manuel checks show everything is OK. Since the car wouldn't present, brought the car home.

A few days later, same hesitation issue only more pronounced hesitation and sputtering with acceleration from a stop. Had to pull over, let the car 'warm up" for 5 minutes, and then was able to drive home with no further issues that day.

A week later, hesitation with severe stumbling happened, pulled over, to let the car "warm up" for 5 minutes. When ready to pull away from the curb, put car in drive, pressed accelerator and the car died. Turned car off, waited few minutes, then back on and tried "warm up" again. Again, when pulling away from curb, car died. Had to tow car to the shop.

Mechanic has had my car for a week now, says they drive my car daily, but car won't present issue. They say the car has to hesitate/stumble when they are driving it in order to diagnose and fix the problem.

Don't want to randomly replace things, but something is causing the problem.
Any thoughts?

Distributor?
Fuel pump?