Mystery hesitation - 2003 civic

I have a 2003 civic (automatic) with 72000 miles on it. It recently began hesitating from a cold start.

It will hesitate once or twice, I stop, and after that it is fine for the rest of the drive. Since it is sporadic, it’s hard for the mechanics to see what is going on.

I took it to one mechanic who cleaned the fuel injectors, cleaned the throttle plates, flushed the transmission fluid, and replaced the spark plugs. He also put a note on the bill that the transmission may need to be rebuilt soon, although he didn’t mention that when he told us what he did.

The car drove fine for about three weeks after that, and then it started hesitating daily.

I took it into another mechanic who was able to see the problem for himself when I left the car overnight. He scanned it for

error codes, but didn’t find anything. He said he checked all connections and wiring and cleaned the wiring but that was all

he could do. He said he thought it was a bad O2 sensor, but because the check engine light isn’t on, he can’t do anything.

Meanwhile, he said I should keep driving it in hopes that it will get worse so they can figure it out. I’m worried that I may be causing long term damage, and that I may get stuck in the middle of an intersection without power.

Should I wait it out and follow their advice? Is there anything they didn’t check that they should have? Should I take it to another mechanic?

An automatic transmission with 72,000 miles?.  Did it ever have any transmission service, hopefully a drain, clean filter and refill with fresh fluid?  I know the manufacturer's don't tend to tell you that the transmission fluid should be change every 30-40,000 miles, but it should.  I certainly can't say that is the problem but it is possible.  

On the other hand I wonder if the throttle sensor may be acting up. IAC?