No. They have already allowed Starlink competitors to launch on Space X rockets.

There are multiple players working on constellations of low-orbit satellites competing with Starlink

While you're technically correct, the parent is more correct. Competitors have to pay normal launch rates. The competitive service needs to include those costs to end users.

Starlink "pays" for launches at cost. While we don't know what SpaceX's cost margins are, they are not trivial. To setup a low orbit constellation is extremely expensive and competitors lose millions per launch that Starlink gets to reinvest.

There's been 136 launches of Falcon 9 for Starlink. ~US$62m per launch? If their margins are 20% that's that's $1.6b in savings. And I bet F9's margins are closer to 50% - supporting Starship and more.

Sure, but Starlink launches are at-cost, which is much, much cheaper than the cost for external customers.

Starlink also has launch priority. Good luck with getting 50 launches a year as a customer...