The primary market works in part because there is a secondary market. It would be a lot harder to raise the initial capital if the initial investors wouldn't have the prospect of unloading their shares to the public. That doesn't mean I buy into the myth of the stock market as an efficient allocator of capital, though. For one, it isn't efficient. For another, if you look at e.g. AAPL a huge amount of capital is sunk into the stock, while the company doesn't know what to do with all the money it is generating.

Post-IPO, companies can also raise additional capital by issuing more shares (either to the public or by e.g. issuing RSUs to employees). Only in those cases is the stock price relevant. Otherwise, the market can be mispricing a stock without any bearing on the functioning of the company.

'It would be a lot harder to raise the initial capital if the initial investors wouldn't have the prospect of unloading their shares to the public."

It wouldn't, because there isn't anything else they can do with the capital other than sit on it in a bank and get no return.

The myth that the stock market is relevant is put about by people who work in stock markets. Much like share buy-backs are the same as dividend payments. Exactly how giving the company's money to people who want to stop investing in the company is the same as giving it to those that do is a triumph of using dodgy maths and statistics to fool people.

Doesn't it give the company higher valuation and higher chances to attract more capital?

Yes. But isn't the IPO most often represent the biggest capital inflow of a company's life typically? How would any post-IPO debt issuance or sales of stock compare to the cash generated from the IPO.

Unless you're Apple, which raised about $100M in its IPO, and currently makes about $500M in revenue per day.

That's not directly tied to their stock price though. Apple could suffer a corporate scandal tomorrow and their stock price could fall 20% and revenues could be largely unaffected.

I just don't follow the reasoning that stock price enables resource allocation.

It doesn't; you're exactly right. It's something people who hold for value (i.e. baseball cards or tulips) rather than for income (i.e. dividends, or what people used to hold stocks for for centuries) tell themselves.

Sort of but not really.

Companies can issue bonds if they want to raise large amounts of capital. Those are traded on a different market and are fundamentally different than stocks. They are debt instruments, not ownership.