Car died at stop. turns over, but won't start

According to what you stated the problem comes from the fuel supply again. The installation was correct fitted ? Check again whether the fuel pump is in a correct place and level. Probably there is a special location for the fuel pump seat. If it is in a wrong level or location on turning any curve the pump is not having a good suction with enough fuel send to the engine and dies. Also check whether after the last night if fuel is coming out or not. If is negative check if the fuel pump is working or not. If is not,check for any fuse or relay. If pump is working but not fuel comes out check : a) do you have enough fuel in the tank?, b) check for a blogged fuel filter, c) broken pipe line.

Mine just had these same symptoms, and it was because the fuel pump had died, and I had to replace it. But first, check the fuse panel to see if maybe there's a dedicated fuse for the fuel pump. If so, it may e as easy and inexpensive as replacing that. I wish you luck! I had no transportation for three weeks before I could get it fixed!

If you have voltage at the fuse and relay, and at the pump, but no fuel pressure, I would say the pump has died or the tank is empty.
Have you checked for voltage at the pump yet ?

what about a broken timing belt
cam shaft /crank shaft position sensor or failed immobilizer

sounds more like the ECM than a relay, to me.

check fuel pump fuse and relay also the fuel injection fuse and ecu fuse, if they are ok remove the back seat and you will see a cover remove the cover and check for voltage at the pump plug and ground, you hear the pump work if not need new pump

Check your battery and alternator connections for tightness. Does it crank over, but won't start? If it won't even crank over, then likely some connection did come loose, like an alternator connection, possibly your fusible link (near the battery).
If it does crank, but won;t start: You'll just have to start from scratch to find the problem. Try to determine if it's fuel or ignition-related. Check for spark at a plug wire. If you have spark, then ignition is not the problem. Check if your fuel pump works. When you turn key on, you should hear the pump buzz on for a couple of seconds. It should also be on when cranking and running, but you can only hear it, or should, when you first turn key to on. If your toyota has a schrader valve on the fuel rail (a valve for checking pressure-it looks like a typical air valve, like on your tire, but is for attaching a fuel pressure gauge) if it has one punch the center with a little screwdriver-gas should shoot out if you have any fuel pressure, If not, have your fuel pressure checked.good luck.

If the engine won't fire up, but your starter turns over, you may want to check your timing unless you have a check engine light on. If the CEL light is on, have the codes scanned.

If the engine won't turn over with the starter or it clicks, have your battery tested at autozone or advanced auto parts. Replace the battery if it is bad. Else, have your starter tested.

If the engine still won't turn over and the battery and starter are tested fine, you may have a blown fuse, bad ignition switch, or a failed relay.

If the engine fires up but randomly dies and starts back up, you either have a failed sensor, failing fuel pump, clogged fuel filter, bad plugs or wires, or a failing ECU.

I’m happy to help further over the phone at https://www.6ya.com/expert/jeremy_69f3cc28d95bf514

TURN ON IGNITION SWITCH LISTEN FOR FUEL PUMP OPERATION.YOU SHOULD HEAR FUEL PUMP RUN FOR 3 SECONDS AND TURN OFF.IF NOT CHECK FUEL PUMP FUSE AND RELAY.IF ALL IS GOOD CODE SCAN FOR PCM FAULT OR THEFT SECURITY FAULT BEFORE BUYING A NEW FUEL PUMP.

Sounds like the fuel pump has an issue with a safety on the pump itself.