GPS watches don't need service, they just need line of site to the GPS satellites. Uploading to Strava requires service, but that can be done any time after the activity.
GPS tries to cover the whole globe, app uses GPS to get location. Ship probably has internet connection in the from of wifi or a cell tower with a starlink or other sattelite backbone link and app's traffic is encrypted so ships firewalls cannot easily block this
Under wartime conditions they would but rules are looser out of combat so sailors can use personal devices for entertainment etc to keep morale up.
Are the sailors really such wimps that they can't go without Netflix?
It's more just that peacetime navy life is extremely boring and repetitive when you're deployed at sea. If you're lucky it's mildly mentally stimulating repetitive boredom but there are people who's whole life for 6+ months at a time involves chipping paint off various pieces of the ship and applying fresh paint in the unending fight against corrosion.