I thought I was downloading them all from the strava website because I figured they would disappear. Unfortunately, when you download them from strava they strip the time element so all I have is GPX tracks. I suppose theoretically you might be able to deduce speed by point density but honestly I don't know.
Did you ever upload that non-timestamped set anywhere? When I grab a timestamped GPX from his Strava account, it appears that the track points come at 1-second intervals. I'm not sure if that's the original recording rate of the TomTom, or if that's a re-generation of the GPX by Strava using their own interpolation, but either way, a first assumption might be that the non-timestamped GPXs use the same 1-second interval. If so, it would be relatively trivial to write a script to re-populate the GPX with timestamps, assuming the track points stay in time order within the file. Such a reconstruction probably wouldn't pass as evidence in a court of law, and I'm not sure how many more almost-smoking guns we need, but I wouldn't mind at least exploring it.
I've seen Strava activities that definitely don't have a point every second. I think that is probably the raw data.