The ECT sensor will not keep the engine from starting. An overheated fuel pump doesn't care if the relay is jumped, listen for the pump priming when you switch the key to posII.

There is just not enough data in your post to make an educated guess.

Keep some ether in the car ( bring inside in hot weather) and diagnose bad fuel delivery with a 3 second shot next time it occurs

From there , you can service fuel delivery

It’s much better on 99-2000 than previous. I’m original fuel pump and relay.

If it’s did not kick on starter fluid that is pointing away from fuel delivery

Time to look at ignition.

Coils, plugs, check voltages

If it fires in ether, and the injectors are delivering fuel, the fuel delivery amount is wrong. I think you need an oscilloscope to confirm the injector pulse ( more accurately , how long the ECU isnpulling the injectors to ground An ECU swap is the Mossberg method if you can’t get a scope , but you...

They are NOT interchangeable and also visualy not the same.
Yes...main fuel injection relay could make problems when it becomes hot....just like any other relay.
Try to swap it with a used one and see if problems go away.

There's also the gray fuel injection main relay up under the radiator - or maybe that relay is not used on a 98? My 98 is not here or I would check. Someone else will remember. It's basically guaranteed to be an electrical/electronic part, that when it heats up the resistance gets too high. V = IR ...