starter makes car start, turn off shortly

No, you can't do that because then the system won't have power.   As far as needing to press buttons, the system itself probably has a type of immobilizer feature where it prevents someone from starting the car while it is armed, but usually that's deactivated when you disarm.   What buttons do you need to push to get it to work?

I looked up a wiring chart on this vehicle but I think mine might have an error.  It lists both "Ignition 2" and "Accessory" as GRAY / YELLOW and it doesn't mention a BLACK/ Light Green wire.  Also, it says that this car does have PATS.   (I made sure I was looking up the T-Bird this time, too.) That isn't consistent with other information I've found or your findings.

Next time you try to remote start the car, try this:  put the key in the ignition, but don't turn it, then activate the remote start.  If it works with the key in, then you must have PATS.  If not, then that isn't your problem.

I don't see how the alarm would know if the key is in the ignition as long as it is in the OFF position, in other words, leave it in the position where you can take it in and out.  Then try remote start.  If it starts OK, then it's PATS.

It would seem to me that, if Ford was using PATS in other vehicles at this time, I don't see why they wouldn't have it in this one.  But, best to check to be sure as you don't want to buy parts you don't need.