Hello, Mi FP4 switches off by itself and does not go on again unless I remove and put back the battery in place. Now even removing the battery and putting it back does not work, no way to restart it…
This happened to me when the phone crashed on incompatible 5G cells. You can try disabling 5G in network settings.
When you remove the battery, try to press the power button as if powering on the phone. This will drain all remaining electricity, which could help with the startup once you insert the battery again.
Now I’m not able to start it up anymore… I’ve tried to press the start button without battery as you suggested, but still no signs of life 
Also when I connect it to the charger the display stays off…
Then it looks like a dead core module You can try another charger, another cable, powerbank… But if it’s dead on all of them, then the phone is probably done…