recommend new Mouse- Logitech can no longer be compatible? (with Catalina)

This article:

https://support.logi.com/hc/en-us/articles/360023202814

says that Logitech's v3.9.8 may not work with Catalina but that 3.9.9 is available.

EDIT: Just found this after upgrading my Logitech software:

https://support.logi.com/hc/en-us/articles/360052872754

and this:

https://support.logi.com/hc/en-us/articles/360053649633

The old Logitech Control Center will not work under Big Sur (MacOS 11.0). Catalina is still good. Looks like they are working on solution to other functions.

No idea where you heard or read that. I'm using a Logitech mouse in Catalina right now. I have two older ones I keep around as spares that also work. In fact, I've never used anything but Logitech mice.

imho, the magic mouse (old iMac) and now the magic mouse 2 (w/ built in rechargeable battery) is the go-to mouse for use with macs. it seems that the gui was designed with the magic mouse in mind. you can do things with it that 3rd party mice simply cannot. i know they are pricey, but what about apple has ever been cheap? quality costs $$. period.