Can Lightroom 3 be used on its own, now the servers are no longer used to verify registration?

Eventually that old program will not run under a new OS, or even perhaps just a OS update. In part this has already occurred for pre subscription Lightroom under MACOS, specifically running 32-bit installers as used by old Lr.

As for just keeping the old OS, that is just begging for some hacker to visit your computer.

Why don't you just try it? I haven't tried using LR 3 in a long time, but it should work on Windows, and maybe on older versions of Mac OS.

(There was no verification until LR 6, and those servers still work)