Android

They should do it. Old phones can be repurposed for stuff like smart mirrors, not having to worry about the battery in that case would be great.

Nowadays, pretty much every phone supports battery idle mode anyways, which is the same thing: They don't trickle charge the battery, they just run off the charger entirely.

If we get replace batteries, implementing this too is a negligible amount of effort.

If your bios is getting reset it's possible your system uses a cmos battery and it is dead. It's separate from the main battery and handles date/time/bios.

Just look for something that looks like a button battery inside your laptop. Sometimes it's wrapped in plastic. My old Dell had it wrapped in rubber which held the wires in place and had a special connector.

Newer systems don't have them - they just rely on the main battery. But if yours is losing time/settings you might have one with a separate battery. It's usually a cr2032 or similar.

Wouldn't using a degraded battery and a constant wired connection already be sufficient for this use case?

You want a landline?