Back story, short(?) version.
1.7 engine was recently rebuilt. New case, pistons, and cylinders. FI and other components brought over with new injectors and a used wiring harness with ECU. After bringing it home I drove it to work, 20 mi each way and it ran fine. When putting it away the spring popped off the accelerator. After fixing that it went for a very short drive with no hiccups. Then set off to work again earlier this week. Started and ran ok (cool morning) but stalled when I had to stop for a school bus. It then would not restart. Turned over but would not start. Towed home. After sitting while at work I put a trickle charger on it to be sure the battery was ok. Again it started ok but when it did stall it did not restart. Pushed into garage and attached trickle charger. Today started it and was going to let it sit and idle for a while. When I checked on it it had stalled and did not want to restart. Let it sit, put trickle charger on to be sure battery was charged, and waited. Went out and it started ok and I sat in the car while it idled. Did ok for a couple of minutes then I had to keep gas on it to keep it running. I let it stall and again it does not want to restart after running a bit.

Any suggestions on what is going on, what to do next, or directions for repairman?

Check the battery. How old is it? That would be where I'd start. Next, I assume the alternator light is not on, but output could be an issue it not the battery. It sure seems that the battery is not holding a charge. It it's because of a weak alternator or battery you will need to determine.

My money is on the battery

if you mean not turning over, that is one thing, if you mean did not start that is another. If it did not start, a quick guess is a bad head temp sensor. Ifit did not start them maybe a bad battery, or a battery not being charged

Battery was new Feb of '18. Car spent Oct of '18 until Aug of this year in the shop with the rebuild. Since it is not driven much I keep a tender on the battery. I will try to meter the battery and alternator this weekend.

When it stalls it will turn over so there is "juice" to run the starter motor, the engine just does not fire.

Cylinder head temp sensor was replaced with rebuild per the invoice. I have included a parts list from rebuild for reference.

Doesn’t sound like the battery is the issue if it turns over.
Is the engine flooding? What do the spark plugs look like? Is the fuel pressure at spec?

Could be a faulty fuel pump.
Mine did something similar. Could drive it and then it would stall and not start. Waiting several minutes to several hours, and it would start up and run but eventually fail again.

Give the man a cookie, it is the fuel pump!