Just stopping the sync on future runs would suffice.
Yeah, that’s an idea in Pause/Stop sync toggle that I do intend to implement.
getting each GPX datafile onto a server
Outside of the advanced LifeMap (which uses mbtiles), there are no files involved in CityStrides at all.
Then why connect at all?
This is a decent question for some people and particularly right now in CityStrides history.
I started with APIs because – going back to day one of CityStrides – I’m a Runkeeper user, they have an API, I’m not going to futz with files if I can collect data automatically through the API.
Things have changed a bit 
since then, and I’m hearing that people are much less manual-effort-dealing-with-files-averse than I am. I expect at some point I’ll offer some kind of “sign in with email/password and manually upload files” option, but right now I’m working on the Garmin integration.