@yfzdude and @dankcushions , thank you for joining the discussion, and glad to have someone who is familiar w/this controller... geez I can't believe all this time that I didn't realize that the Logitech logo was a button as well(!!). I may have to do that controller test again to see what it comes up as, because somehow I missed it even there. So fortunately, I was able to assign it as the HotKey button, although it didn't actually show up the first couple of times I pressed it, which isn't entirely reassuring, but it seems to be working now.
So that said, I may wanna look into getting different controllers moving forward (including possibly that Dualshock that you mentioned), but for the moment this seems to get the job done and pause the game as I'd originally hoped, so yeah, doesn't look like I may wanna return this controller as I'd feared.
But yeah, re: that weird "Mode" button, you're right, it DOES seem to alternate you btwn/enable and disable the cross-shaped directional controls and the thumbstick one, and there'd been a few times during games when one control didn't seem to be working (when I'd pressed Mode without realizing it), and had to switch to the other directional option. Usually I'd end up just resetting the game to get it back to normal, but glad to know now what the purpose of it is.