Is it possible to play Quake with a Logitech controller or is this strictly mouse and keyboard?

And presumably, you mean Quake 1, without any source ports?

I Take it you mean without a mod it can't be played with a controller?

No. It may not have been explicitly designed around it, but you may need to (if capable) switch to dinput mode.

And they're not mods, they're simply repackaging of the Quake engine into something that's less...90s tech.

That mean they remake mods.

technically, i thnk gamepad is possible. Such resources as PCGW is not a good source for answering this question. But not expect analog control. Proper analog control. Use sticks can lead to some absolutely differ things even if you reconfigure controls - and for dinput as in old time you always should do that.
D-pad is same as play game with no mouse. It be optional back in 1st half of '90. At the time of Quake 2 it be normal to play games in many ways, not only gamepads in shooters.