What do you prefer for snagging a few tasty walleyes, or whatever will bite through lake ice? Why?

Rod and reel. Tip ups are boring unless you're on a hot weed bed full of northerns.

I've had better luck with a rod and a fish finder (flasher) than with tip-ups although I remember fishing Wyoming with 6 lines per man.
5 tip-ups with minnows and a rod and flasher, The ice looked like a mine field of drilled holes when there were 3-4 guys in the same area.

Rod/flasher combo puts a lot more fish on the ice for me.

Jim