I don't know what to say.
Besides this '98, I had a '99 (not my current '99) that I was working on a few years ago and the only key physically broke. A 3rd-party locksmith with their cloned knock-off keys didn't fix it. After getting a new key from them, it briefly started, then stopped due to lack of fuel. Cranked, but wouldn't start. No fuel pump priming. After taking it to the dealer for new keys, it worked again, and the fuel pump primed when you Key-On/Engine-Off - the same behavior this '98 did, after we resolved the PATS issue. Literally as soon as the new keys got programmed, the fuel pump started priming and the thing started.
Also, my sister used to have an '02 Mustang that she was accenting in gaudy pink. She bought some pink knock-off PATS keys for it off of eBay that would never program correctly, even from the dealer. Keys were cut, mechanically functioned, but trying to start it resulted in the same behavior - no fuel pump prime/operate.
Eminently replicable, we could put the OEM key back into it, start it up and it runs. Taking that key out, and using one of the knock-offs, it would briefly start with what fuel was already primed in the system, then die when that ran out.
I attempted to pull the "chip" out of her single working factory key and install it in the pink key, and I broke the damned thing. After buying her replacement STRATTEC keys and having those cut/programmed, starting restored, and again, with the "real" key, you can Key-On/Engine-Off and hear the pump prime, but with the knock-off keys, that behavior didn't happen.
I really don't know what to say, other than my direct experience with (3) different Fords (two Explorer, one Mustang) is that a PATS failure disabled the fuel pump, and resolving the PATS issue re-enabled it.
tl;dr - The fuel pump makes a very distinct sound when it's activated to prime with the key on. On 3 different Fords, with PATS faulting, it did not activate/prime. Resolving the PATS issue, and the pump activates/primes.
Edited to add: There are/were no aftermarket systems on either of the Explorers that had this issue - bone stock. The Mustang did have an aftermarket alarm on it, but it was just a cheapy basic "woop-woop" system, it did not have any advanced features like a component disable or anything like that.