SpaceX is a space exploitation business, Starlink being the foremost example but also commercial and governmental launches of Falcon 9 and eventually Starship. Even going to Mars is ultimately a mission of exploitation, not exploration.
Space exploration is the duty of governmental space agencies such as NASA, who (assuming sufficient budgeting) can all literally afford to run red ink for entire projects and not have to give a damn.
Starlink was basically created to get SpaceX's launch cadence up. Which it absolutely succeeded at. SpaceX exists to cause space exploration/colonization/all-activities to occur, specifically going to Mars, but also more generally. Which again, it has absolutely succeeded at.
NASA and other space agencies are indeed picking the missions, but SpaceX has been a huge enabler here.
* Whoever gets to Mars (and the Moon for that matter) first in a permanent fashion gets to write all the rules. Full stop. It's also why the US really does not want China achieving a Moon presence first.
* Starlink is competing (and winning) against all the incumbent ISPs for being pieces of shit one way or another, especially incumbent satellite ISPs like Hughesnet who are their immediate competitors.
Space exploration is the duty of governmental space agencies such as NASA, who (assuming sufficient budgeting) can all literally afford to run red ink for entire projects and not have to give a damn.