The problem does suggest a fuel mixture error.

A component critical to fuel mixture that is somewhat prone to failure is the "oxygen sensor" that detects unburned fuel in the exhaust gas to tell the computer to make the mix leaner. (More air - less fuel.) Many recent vehicles have two oxy sensors, with one near the actual engine exhaust point and a second one nearer the catalytic converter outlet.

Since a richer mixture is needed for cold starting, the oxy sensor is likely to pick up an excess of fuel during starting. As the engine warms up and the mixture is "leaned" the excess has to be burned off, so the sensor actually lags the actual exhaust gas state as the mix leans, if a fully rich mixture was used for the cold start.

Residual "excess" on the sensor, when you try to start a warm engine, may make the computer think the mix is too rich, so it provides a very lean mix to the engine, with insufficient fuel for the restart.

If the engine runs long enough to "purge" the oxy sensor so that it gives a correct reading for the current state of the engine, the mix will be right for starting. If the engine didn't run long enough to clean up the excess from the last cold start, the mix will be too lean for re-starting.

A good sensor should clean up very quickly, but one that's been in service for a while may have a much slower response. Even a sensor that's gone completely "dead" probably won't prevent starting under most conditions, and a dead or dying sensor is usually seen just as a degraded performance while the engine is running. The sensor also can be "killed" rather quickly by fuel contaminated with anything that doesn't "burn out" quickly, with even small amounts of leaded fuel being a most common cause before it was mostly eliminated from the market. Some aftermarket "fuel additives" still available can have very negative effects.

Exhaust system leaks, or clogging, can significantly affect how quickly the sensor cleans up, and how closely it follows actual mix conditions. Exhaust line problems that persist for too long can actually cause permanent sensor failure. An inefficient (worn out) catalytic converter might allow the secondary oxy sensor to call for leaner mix.

Some engine computers from around the era of the vehicle in question did not record an error that directly indicates oxy sensor failure, although I can't say whether that's the case with the specific vehicle involved. The diagnostic test for sensor condition is a simple resistance (any VOM/multimeter) check, but resistance values have to be matched to actual sensor temperature and you'd need specs for the specific sensor on the vehicle.

This is only one of many possibilities; but it's one commonly mis-diagnosed and ignored around the time that the vehicle in question was new. (New oxy sensor - $20, "carburetor rebuild" or mixture control module replace - $400, hence a bit of bias in what the mechanic tried first)

John