Subsidies for historic populations without price controls would be a windfall for landlords, and an accelerant to runaway general unaffordability. You could probably split the difference with weaker rent controls and subsidies that make it look like stronger rent control from the renters side, which might mitigate the worst problems of either policies (albeit, at thr cost of combining thr problems of both.)
What would probably be better is to find a way to lean into displacement, or at least accept it, while giving displaced residents a stake in the unleashed value.
Strong rent control but with a buyout option where the the bought-out renter gets some share of the excess (compared to what they could have been charged with rent control) rent over a specified period with some floor might be an option.