The EU did do that, decades ago. The problem is that it requires constant investment. It's not profitable. The governments helped build it, abandoned the companies until they went bankrupt, rescued them (they're not actually insane enough to just abandon working satellites), privatized them, they went bankrupt, ...

Obviously the satellites were never modernized. But it does work, for a few thousand terminals for all of Europe with 2x to 10x the ping Starlink provides.

It's like a lot of things in the EU: on the one hand the EU absolutely requires this infrastructure, or they become dependent on foreign nations for critical infrastructure. But they won't pay. It's not even that expensive. Starlink was built with budgets that would be double-digit millions per year per EU country. But the main problem always repeats: they can't agree who gets the money/business.

If you calculate the lifespan and cost of a Starlink satellite you will come to the obvious conclusion: it will be very hard for Starlink to break even. Of course, the same can be said for most of Musk's businesses (perhaps all. I'm not actually aware of any exceptions)