“rebooting” is not a thing ova here. yeah, you can accomplish that but that’s not what you want. utilising something like InputRemapper + e.g. Plasma shortcuts, you can launch a big-picture UI, like steam or plama-bigscreen when it’s ready or somesuch, when you press a key combo on the controller or mouse or keyboard or any combination thereof.
holup - you shut down the laptop and in such a state it drains the battery?! I mean, that’s so outside of the OS’ functionality, it don’t matter which one you got. the only sensible conclusion is that shutting down the laptop in debian doesn’t turn it off, there are no other explanations.
fedora is more modern by way of kernels and DEs and whatnot, but I’ve looked up your hardware, that’s an 8th gen i5/i7, that’s plently supported even in old bookworm.
one thing to lookup is in BIOS, my T480s (same generation) had a power management setting in BIOS that was either Windows or Linux, so make sure yours is set correctly.