In case you haven't surmised from the answer, yes, the GPS receiver is independent of wifi, though when you have both, they can work together. If you're not interested in seeing your progress on a map while you are riding, the tablet can probably do what you want.

One alternative to consider is an inexpensive GPS such as the Garmin Edge 200. It's more than $75, though. I like it fairly well. I press Start, and it records my GPS coordinates over time. Then when I connect to a computer at home or work, it plots the GPS coordinates over the map. It uploads my ride to the web and provides quite a lot of statistics and graphs. Here is one of my rides. The unit is a little bigger than a watch, so it fits in my pocket if I don't want to mount it on the handlebar.

Another alternative is a basic bike computer which doesn't use GPS. You can get them for $6 to $50. I find the cheap Chinese models work just fine. They give speed, average speed, distance, and other stats. Since you want to keep yours in a bum bag, I guess you'll need a wireless model. I think they start at $15. But you'd have to move the sensor/sender from bike to bike. Normally, it attaches to your fork and senses a spoke-mounted magnet.