I have to admit, I'm curious as to how you were able to determine that it would start running if you got it rolling first.

I've never heard of such a phenomenon, but in these days of computer technology, nothing is out of the question anymore. I would treat this as a "no start" and diagnose it as such. With the vehicle stationary.

First checks: fuel pressure, scan PCM for trouble codes, check for presence of spark at the spark plugs.

OK, that's good. The only other point here is: are you the original owner? There may be some aftermarket electronics stuffed up under the dash that you're not aware of. If this were in my shop, I'd at least peek under there with a flashlight and just make sure.

The flashing odometer is what points me in that direction, that is an odd symptom. I'm inclined to say that you have a general power failure of some sort, like a battery cable, ground cable connection, a bad fuse that is backfeeding power in some weird way.... I would go through all of those basics: check all fuses, check and clean the battery cables as necessary, check the battery cable connections at the engine and fuse block.

I don't know that I would replace any more parts on this until I had a scan tool in hand to see what is going on when it won't start. There is potentially a lot of data to be gained from a good scan tool. This is one of those concerns that could literally be just about anything, from a bad connection to a bad computer. Scan tool data will probably hold the key.