You're not accounting for the transfer of cash out of the company. If you have a company with a business worth a billion dollars and a billion dollars in cash, it should have a market cap of two billion dollars. If it uses half a billion dollars to buy back shares then it should have a market cap of $1.5B because it has $500M less cash. Then it also has 75% as many outstanding shares, so the value of each share is the same.

But the buyback often increases the value of the shares because it allows investors to express their preferences better. If the company has a $1B business and $1B in cash, there is no option to invest in only the business, only the combined business+cash entity. If you think the business will give 9% returns and the cash 2% returns and you have an alternative investment that gives 6% returns, you won't invest in that company. But if they separate the cash from the business then you're willing to invest in the business, which makes the business worth more to investors and increases its share price.