The common practice of immediately selling whatever percent of shares is required to pay taxes on them is something that employees are choosing to do, supported by the trading firms that help implement vesting schedules and stock sales. Employees are allowed to keep all of their shares and pay tax on whatever next interval is required instead, if they wish. One can only follow this practice of selling shares immediately to cover tax if the company's shares are liquid, i.e., the company is a publicly-traded company with an IPO.
I suppose in theory one could receive stock in a private company, and sell shares on the secondary market to cover taxes, but with private companies you can't take it for granted that (i) you'll be allowed to do that at all, or that (ii) there will be a buyer for those shares at all, or at a price you're happy with. With a publicly traded company, it is taken for granted that there's always a buyer for the shares, and at a price that is commonly known and accepted.