I have this same issue, even after following all the steps above.
I just bought a brand new MacBook to replace my 2012 MacBook so that I could start using Final Cut Pro for video editing (I had maxed out what I could do with iMovie with a couple of my side projects). My iPhone 12 Pro Max sometimes had issues pairing with my MacBook for Keynote, but my iPad (2020) never did. This was annoying, but generally fine because I usually use my iPad while teaching for controlling the Keynotes, since I can use the Apple Pencil (the only reason I would try to use the iPhone would be if I accidentally left the iPad at home; unfortunately, it's not been a reliable backup for that option since the Phone and Keynote don't seem to want to talk to each other on a regular basis).
However, now that I have my nice shiny new MacBook (Monterey 12.1), neither device can be used as a Keynote remote with this new MacBook. Apparently, I'm going to have to go back to my 2012 MacBook (High Sierra) to be able to use my iPad to aid in my teaching?
I can still Bluetooth connect to my 2012 MacBook, if necessary, too. What's up with that feature not working with my iPhone or iPad on the new MacBook, either (same as with Keynote- I can get the 4 digit pair code, and then I get an error from the Bluetooth driver saying the devices aren't compatible when I try to connect)? Considering I just spent several thousand dollars to get this computer, I'm not happy that such basic features of the operating system and integrated software of these devices are working on an old, outdated MacBook, but not the most current and up-to-date one. What happened to "It just works"?