Usually what they have is rent stabilization rather than control. It is tricky business to really estimate what is reasonable or not under that. For example, rents in my city are allegedly "decreasing," yet if you live in a rent stabilized unit, your landlord is probably incentivized to increase your rent to the limits of the ordinance, it could be 4 or 5% so not unsubstantial especially it being compounded by the year.

And what is more, when a tenant leaves and the landlord puts the unit up for rent again, there is no price limit for what they can ask for new rent. They are free to ask above market rent if they want.

So really I would not say the burden is shifted entirely on the owner of the building when they still possess two levers for ameliorating overhead increases: increasing to the limit of the RSO even when the market doesn't support any increase, and increasing rent on the next tenant. In high demand cities the market rate is not so much a wall, and landlord generally enjoys good success pricing above market rate and still leasing the unit out in a reasonable timeframe for them.