You are 100% correct except for
though it seems to be failing at that goal.
Starlink is expected to bring in 2 billion In profit this year, it is successful https://arstechnica.com/space/2025/02/starlink-profit-growing-rapidly-as-it-faces-a-moment-of-promise-and-peril/
Starlink is bankrolling starlink, it's expected to generate 12 billion In revenue this year with 2 billion of that being profit
Before it paid for itself spacex did and still does have a lot of very wealthy private investors willing to throw significant funds at the company