A particular MM might lay off the call risk by purchasing a call, but all they're doing is passing the risk on to a different MM.
There aren't that many natural vol sellers (hedge funds, mostly? and implicitly the Fed), so MMs as a group have the function of converting implied vol into realized vol via their delta hedging activities.
shares are just long dated calls. There are plenty of routine covered call writers running the wheel from wealth managers to boomers like my dad.
But you are right that to you need to have the capital to face early assignment if you are short and the capital to take delivery if you are long volatility and liquidity is low relatively.
There aren't that many natural vol sellers (hedge funds, mostly? and implicitly the Fed), so MMs as a group have the function of converting implied vol into realized vol via their delta hedging activities.