Of course they do. It takes 2 days for a stock trade to settle. They HAVE to buy short to sell it to you instantaneously. But, they are not shorting the stock as suggested - it is the simple matter that they could not get you your shares any other way. Any stock you get in a trade is not settled for 2 days, so it is short during that time. As far as you are concerned, you have that stock, but it is not a done deal yet - for 2 days. But you are allowed to turn around and sell it anyway - the MM's are there to allow you to buy and sell at will.
BUT - they don't 'short' stocks for themselves - they are not allowed to be traders, they are more like brokers, but not exactly the same. They are facilitators, they ensure that the market stays liquid, and they make sure that you can buy and sell at will, assuming you have prices that they can achieve for you. But MM's don't accumulate shares for themselves - they accumulate them for buyers that have placed orders.
It is large shareholders that control the prices, and make a mint doing it. Not the MM's, who make their margin on each trade.
MM's are the middlemen. They make something on each buy or sell, but they don't control the price. Shareholders do that.
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