Kinda need the full text and formatted version of that since you know what you're talking about and nobody else does.
One thing to do is hook up an OBDII scan tool capable of live data, look at what it's displaying and look for pending fault codes.
So to confirm, you replaced one or both head gaskets and did a compression test afterwards and the numbers were good, all cylinders within a small % of each other?
Have you tried spraying starting fluid into the intake to see if it'll start and run on that? That might indicate that the erratic fuel pressure is causing a delivery problem. It makes me wonder if there's some fuel injector cloggage going on since you report it holds pressure for 30 min which should indicate that the pressure regulator is good.
Another thing to keep in mind is the history of the vehicle. They said it needed head gaskets but how did they know? Did the oil look good when you bought it, did you take it for a test drive THEN look at the oil? I'm wondering if they had engine damage, changed the oil to hide it, you looked at the oil and saw clean oil because they changed the oil, then drove it but didn't look at the oil after the test drive?
I'm not placing blame, just saying that the engine could have a lot of miles with coolant in the oil and it's not just some component "X" that is the problem but rather that the engine is worn out. If you can get it running more than a few seconds you should be able to tell based on how noisy it is.
Wait... you wrote "got it home". Do you mean that it started okay, that you drove it and it seemed to run somewhat okay, all the way home then this problem started? That is a more curious situation.
I may be overlooking something or just not know what's wrong, but it could be something as simple as a vac leak. Spraying the starter fluid in and seeing an improvement would tend to indicate either that or a fuel delivery problem.