I've used Pis and Arduinos at work, using them to devise a solution here is way overkill. Plenty of commercial examples can do exactly what I want cheaper and in a much smaller packaging.

As to the phone, I do generally use one in my back jersey pocket to record the track, but leaving the screen on all day kills the battery too fast, and pulling it out becomes cumbersome. Speed is fun to know, but what I really like being able to know is distance to turns and such, for better navigation.