Musk is a savvy businessman before anything else. He would prefer to not have competition.

Not true as exemplified by his Tesla company releasing the patents to their technology.

He knows he can't do it alone. He's elevating the bar for everybody.

By fostering a larger talent pool competition insures a dynamic long term future, lack of it it often leads to creative stagnation. This is the case in natural systems as well.

Don't you think they would get to Mars quicker if they cooperated and shared ideas and technologies instead?

Cooperation between parties with differing goals and management styles results in design (and subsequent dismissal) by committee, whereas competition unhindered by interference from the motivated parties results in landing humans on the Moon.

Yes, if everyone were altruistically motivated to get humans to Mars, we would all contribute our dollars and work together to make it happen.

But in the real world, a lot of people who don't care beans about Mars can still be motivated to invest in the effort by the prospect of investment gains. And those gains require for-profit competition.