No other home button will return touch ID functionality other than the original home button that came with your phone. Does the button look damaged? If not, it could be the cable or connector that plugs into the home button is damaged. This isnt usually the case, but it’s worth a shot to remove the original home button from the screen and install it on a new screen to test. If that does not work, the only option to regain touch ID functionality it to take it to an Apple store. In the past, they have been able to replace the home button and reprogram it to restore touch ID functionality.
Rice doesn’t work. It’s a myth. You need to stop trying to turn it on and off. This can cause irreparable damage. You need to open the phone and properly dry it out and clean it with isopropyl alcohol. If you are uncomfortable doing this, find a reputable shop to properly dry it out.
You are on the right track, but remember the logic board has to relay the touch info to the processor and the screen is plugged into the logic board. If the board took a big impact and any traces were broken during the impact, then you may be out of luck. I don’t micro solder so I don’t know if there are any fuses etc that can go out or need to be replaced, but your issue is more than likely a board-level issue. Id find someone reputable to send it to.
In all fairness, your device is 10 years old. Are you certain these apps are compatible with your version of Android OS? Newer cheap Android tablets are available for 30 and 40 bucks now. I dont jump to buy new too often, but in this case you’d be better off getting something else.
Definitely replace the battery first. If this does not solve the issue, try a restore to factory. If all else fails, it could be a board-level issue. I have seen this behavior for the Note 4 and 5 as well.