> Buying your own stock is like returning money to shareholders.
Right. Before, the shareholders have shares. Afterwards, they have shares and money. The total value has not changed at the moment of the operation (ignoring taxes, assuming infinity liquidity, that the information about the transaction was incorporated already in prices, etc.).
> It's effectively a dividend, by raising the stock price by the same amount they use to buy back stock.
It's effectively a dividend, but it's different in that the share price remains constant for a buyback and goes down for a dividend (again, in the ideal case).
With a dividend: Before, the shareholders have X shares. Afterwards, they have money and X shares (which have a price lower than before, to keep the total constant).
With a buyback: Before, the shareholders have X shares. Afterwards, they have money and Y shares (which have the same price as before, but there are less of them to keep the total constant).
> Like if they use $1 billion to buy back stock, the aggregate value of the outstanding stock should go up by $1 billion.
Nonsense. It's the other way around: the aggregate value of the outstanding stock should go down by $1 billion.
Where "should" applies only in the ideal case, sure, but there is no way at all to construct an argument where it "should go up by $1 billion".