SpaceX is a business controlled by a single man that is really interested in making humanity multi-planetary by building a self-sustaining base on Mars.
Maybe. As part of the control he currently has, I think we can safely assume he has been filling the company with employees who are also very jazzed about going to Mars / making humans multi-planetary. So it really depends on where the power lies when Musk dies.
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.
SpaceX makes sense as a business in the way a mega-yacht makes sense as a ship. The valuation was set by a vanity investment by the Saudi sovereign wealth fund. 2.7 million subscribers can't keep 4500 satellites in orbit and replaced every 5 years. It is a prestige investment.
SpaceX is cash flow positive despite spending multiple billions each year on Starship and Starlink. The only way this is possible is if Starlink is profitable, and significantly so.
If you assume a very conservative $100/m subscription for every Starlink customer, they're making $3.6B a year already and it seems like they can do a lot more capacity.
Edit: just googled and they're predicting $6.6B in revenue for 2024.
SpaceX is a business, SpaceX doesn't care about the Moon because there are no customers interested in going to the Moon.
If market forces shift and companies start wanting to go to the Moon, you bet SpaceX will care about the Moon because there's money to be made.