You say "traditional", I think you mean "designated" (or "specialist"). Designated market makers also received certain privileges, but in turn are obligated to maintain an orderly market.

Historically, not all market makers were designated market makers.

Some HFTs (e.g. some branches of Goldman) do act as designated market makers, receive these privileges, and are obliged to continue trading. Others do not. Same as always, just with machines replacing people.