There is a difference between the CPE (what you use to get the service) and what SpaceX uses to sink your traffic into the global internet. And in fact there is exactly zero incentive for somebody other than SpaceX to run the ground stations.

On the other hand it is completely possible that SpaceX's ground stations will use HW that is similar to their CPE box and thus they can build ridiculous amount of such ground stations (ie. at every IXP or so, althought the "lightweith non-IP transport" proclamation somehow precludes doing that)