How can a GPS with damaged screen be put to use?
I'd figure out the key combinations to start and stop recording, and enable .gpx track recording.

You could then plug it into a computer and extract the .gpx files, and then use tools to play with them:


http://www.gpsvisualizer.com/ will let you upload the file and view it on a map
http://gpxdataanalyzer.com/ is a site I wrote that will analyze the file and return some analysis on it if you pick an activity (Flying, Driving, Hiking, etc.) I should probably add bicycling.
Any number of tools (including the Garmin software that comes with their GPS units) that will allow you to visualize the data 'off device'.