SpaceX has a huge price advantage on launches so a competitor would have to be able to significantly undercut them on the cost of the hardware just to become competetive.
Starlink is promising well under a million per satellite, and OneWeb is reportedly already at a million. Satellite costs might end up lower than launch costs, and even if they're higher that doesn't seem like they'll dominate.
That may be true for heavy ones that are only ever built once but these are light and mass produced. I cannot find the source but I'm pretty sure elon musk said that the rocket costs more to build than all of the satellites on a launch.