Purchasing common stock after an offering (I.e. on the open market) doesn’t do anything to support a company, either. It’s essentially just paper being traded back and forth.
But having the ability to liquidize their stake at almost any time makes it easier for investors to put money into the company, and therefore gets more investors on board.
Unless the company needs to sell more stock or gives options/grants to employees. In either case, a higher price reduces the fraction of the company needed to be sold to get the same amount of cash. An increase is share price is almost always still beneficial to the company.