If it starts up after its initial burp, it's likely not the TFI then.
On the grey ones, there's a terminal that differentiates starting vs running - that's the angle I was taking before.
In your case, what happens if you prime the pump a couple of times (cycle the key on and off, pausing for 5 seconds when turned on) before ever trying to start it?
If this helps, that indicates a fuel pressure issue, which is often a weak check valve in the pump assembly.
Unless it's bothersome, I personally would not sweat it so long as the pump continues to function and keep pressure. The check valve really has nothing to do with the diaphram and motor assembly - there have been new pumps installed that did the same darn thing (it's kinda prevalent on 94-95's for some stupid reason - they use the same pumps we do).
I will say that normally with a pump like that, if you dont cycle the key, normally you'd have to crank for a few seconds but once it caught (started), it would stay running. It's strange that it catches, dies and restarts immediately.
The startup & immediately die, restart and it runs, is a IAB problem. Mine's doing it now along with some of the other IAB problems that are common to 5.0's. For cures, see the Surging Idle tech note in the "Useful Technical Thread Index" sticky at the top of the forum.