Some cities do. Denver has Denver B-cycle, and they are giving out 5280 free annual passes this year (not sure if they do this every year). The non-free option is still a fraction of the costs of the for-profit private companies. There are also a plethora of e-scooter brands though and I see them more often than bikes.

Boston has Bluebikes (e: seems like it's a partnership with Motivate?), also pretty cheap. A lot of people I knew in Boston had some kind of discounted rate through their employer where they got unlimited complimentary 30-minute rides.