is there something like fake battery or replacement?

My Nexus 5 phone would shutdown seconds after starting to boot up. Suspecting the battery, I replaced it with a new one. There was no change, so the battery wasn't at fault. I went out and purchased a Pixel 3, solving the problem.

IDK about a fake battery, but you certainly can buy replacement batteries for most phones, including cheap ones from China. Even a cheap, China one would suffice to get it to boot up. Assuming that it is a battery problem. Hard to tell what goes on with the battery interface, how it's monitored, etc. It's certainly possible that it won't boot without a battery, with just USB power. Ebay would have cheap ones.

Cheap batteries from HCina tend to be C grade. That means used batteries - they test them and anything that holds a charge gets rewrapped & sold. Not surprisingly some are defective, some good.

NT

maybe you could connect a capacitor to the battrey terminals and a resistor to the temp sensor terminals

may or may not work but seems unlikely to damage the phone.