Stock markets help allocate resources to the companies that will make good use of it. We can't say something similar about crypto until speculation stops being 99.9% of the use case and it starts being used for something more practical. We may get to that point, but that's still not a certainty.

… Do they? After an IPO, how is the glorified trading card game of stocks actually helping the companies themselves?

By making stock grants valuable and thus aligning the incentives of the employees with the company?

In many cases, shareholders are represented by a board of directors, to whom the CEO is beholden. Also, executives often see much of their compensation via equity in some form.

Companies can issue stock (whether through employee grants or secondary offerings). They can also buy other companies with stock.

Plus the value can effectively act as a form of collateral for loans explicitly or otherwise.

The dream.

Maybe not ban, but heavy regulation and taxation sounds good to me. Too much of our economy is based around ensuring high numbers for the DJIA and Nasdaq, which is not healthy for society.

I can agree to that.