Following the advice on this Reddit thread
https://www.reddit.com/r/logitech/comments/qhk83k/mx_master_3_not_working_after_monterey_update_on/
I installed Logitech options+ and that fixed my problem.
https://www.logitech.com/en-us/software/logi-options-plus.html#join-beta
I granted the + app all the permissions that it asked for (2 of them) and did not have to reset my 2021 MBP 14'', scroll wheel started working immediately. I'm using a Logitech M275 mouse connected via a usb dongle.
MoMBP14in2021
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MacBook Pro (14-inch, 2021)
M1 Pro
12.0.1
Same issue with my Logitech Mouse M275 mouse. Erratically stops working. Only a system reset temporarily fixes it. Super annoying.