I know that "There are other ways to hedge price risk other than buying the underlying. It's not uncommon but it doesn't "always" happen". "You can buy/sell other options to reduce price risk, or be long a different underlying, or have a natural long/short position (e.g. refiners are naturally short crude oil and naturally long products like gasoline)".
None the less, assume that you bought 1 call and in your case, the MM delta-hedges by buying the underlying asset (stock in this case).
Case 1. $S_0$ (current stock price) $> K$ (strike price) $+ c$ (call premium). Presume you sell your option before expiration.
Then you profit and the MM loses money. To unwind the delta hedge, now the MM must sell its underlying stock at a higher price. It's unclear if the MM profited. I don't know if the MM's profit from selling the stock > MM's loss from buying your call, or vice versa.
Case 2. $S_0 ≤ K + c$. Presume you let your call expire worthless.
Then you lost your call premium that the MM earned. To unwind the delta hedge, now the MM must sell its underlying stock at a lower price. Again it's unclear if the MM profited. I don't know if MM's profit from selling you the call > MM's loss on selling the stock, or vice versa.
In both cases, the MM can lose money! Correct? If so, why would MM ever delta-hedge?