Just having a trace of your route probably isn't very useful in a lot of cases, since it's going to be mostly on roads that are already mapped. Instead, you'd probably want to map the actual paths, in which case you can view your GPS trace in the editor to map from without having to share it publicly.

The Strava heatmap is also licensed for usage in OSM, and in a lot of areas it has pretty good coverage of bike paths already (https://wiki.openstreetmap.org/wiki/Strava).