Tesla is profitable and SpaceX is currently pouring all their resources into two futuristic megaprojects with high upfront costs, so it's understandable that they aren't "solidly profitable" at the moment.
Tesla made 5B in profit last year and is on track to double that number this year.
SpaceX is investing heavily in two projects (Starlink and Starship), both of which swallow huge amounts of money.
They are definitely not profitable now, but by choice.
Is making profit the same as being in a solidly profitable state?
Source for the SpaceX profitability? They raised a lot of money last year, which wouldn't be necessary for a profitable company. Falcon is probably profitable, but it only grosses a couple billion a year. OTOH, Starlink and Starship are both $10B+ projects that aren't generating significant revenue yet.
I have no idea where you'd get the notion that profitable companies don't need to borrow money for large capital investments.
He didn't borrow money. They issued stock. Significantly, this reduced Elon's share below 50% which I doubt he would have done lightly.
Did he retain 51% of voting rights? A lot of people (e.g. Brin, Zuckerberg) own way less than 50% of their companies but still maintain control.
Cash flow != profitability. You can be profitable but have negative cash flow. If you have negative cash flow, you need to borrow capital to keep growing.