Power Off is simple to handle because there's a functioning OS that can be (re)programmed to deal with whatever IR or RF keyboard codes the remote sends to run a shutdown. Power On requires something to always be powered and listening/waiting for a specific IR code; that wakes the internal power circuits to (re)start the device. And yes, about 95%+ of all hardware used for LE (all RPis, all NUCs, etc.) do not have that little extra circuit that remains powered to switch things on again. RPi devices can be modified with an add-on board that hosts an IR receiver and a small power circuit that's always-on and provides board power via the GPIO pins. There are IR/remote add-on "lids" for some NUC devices too; but rather expensive.
TL/DR; If the Beelink box has an IR wake function? you might be able to get a programmable IR remote and configure it to wake the box. I'm quite sure the Minix remote is not programmable though, so even if it does have one you're not going to make that combo work.