Keyboard won't work in recovery mode.

I’ll assume you’ve gone through the pairing sequence?

Does this keyboard work elsewhere, if you’ve tested that?

It’s seemingly either failed, or is oddly incompatible.

i’d return this one, and find a different keyboard.

There are lots of choices, Apple and otherwise.

All that written, Logitech might be indirectly confirming a problem with the following:

https://support.logi.com/hc/en-us/articles/360034762774-Getting-Started-MX-Keys

even if you connect the USB adapter, it won't work; or connecting it via usb A to usb c cable (keyboard charching port) or even usb c to usb c, is just won't work

no updates available

i will try this

https://mackeeper.com/blog/how-to-fix-recovery-mode-not-working/

tomaslinlithgow Said:

"Keyboard won't work in recovery mode.: Hi there, I'm currently trying to enable system extensions and when It asks for my password I'm unable to type anything in. I'm using a Logitech MX keys that is connected by usb-c to my Mac mini M1 and nothing happens, surely I don't need to purchase an apple keyboard to fix this issue do I? Thanks for your time."

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Reset the NVRAM:

Sometimes changes (i.e. updates) are made to the system and then settings (i.e. for keyboard) become manipulated, technically.  So, reset both the NVRAM on your Mac. As for connecting it, you may need this adapter, if the keyboard has not USB-C plug in it USB-C to USB Adapter - Apple

Check with Logitech support (looking for a firmware upgrade roo, if any), as it would seem something odd with that keyboard.

The Logitech MX Keys also seems to be a wireless keyboard and not wired, too.

ps: Above recommendation to clear NVRAM is not applicable to M1.