This shit never works out as simply as you expect it to. With any public policy change there are layers and nuance. If you restrict rent increases then every year you’re gonna see the majority of landlords including the ones who do not increase rent yearly suddenly change course to always up the rent 5% per year, every year. Otherwise as a member of the ownership class who doesn’t increase rent your are losing out on equity compared to those who do. Compound interest never catches up.
If you want to make renting profitable and homeownership obtainable for nearly everyone you need to setup a system where renters gain a share of ownership proportional to the money they put in. Make it so ALL renters eventually own the place they rent if they stay long enough. Ownership increases for every dollar spent towards rent. Evaporate the double dip appreciation for landlords and punish the income so that you gain money but at a slower rate than an index fund.
You also need subsidized lending for individuals who own no home. Possibly make it so 100% of rental income tax goes into a fund that provides mortgages to someone purchasing a primary residence when they do not own any current properties or are selling their primary residence to purchase another one (e.g. moving, not hoarding housing.) Make the application process for these mortgages prioritize families without any owned homes in their direct lineage (e.g. those with parents and children which do not own homes get a higher priority than the kids of wealthy homeowners.)
Something like this can bring homeownership to the masses and bring prices down as investment income from housing is no longer as valuable as other investments. Prices may still stay high, but subsidized mortgages would keep monthly payments affordable for those who qualify while minimizing profitability for hoarders.
Finally if you are still in love with rent control, then do it concurrently with a program like the above. Balance the scales.