I just recently replaced my sending unit, and my car starts the first time around, and stays on but then like you were saying it turns off once it warms off and thas the end of it for basically the rest of the day. The temp sensor being the problem seems reasonable except for in my case it would have to have occured out of nowhere becuase the car was fine before.
I’ve recently started using Sunoco’s Ultra 94 and my problem seems to have disappeared. Like it will still nearly “die” but it always catches on and holds. The 94 might not have solved the problem, it might have just made the problem a bit smaller. I’m still going with the temp. sensor as the part to fix.
Try the Sunoco 94 and see what it does, but at ~1.15$ a litre it might not fly too well with your wallet