Kicking grandma out of her rented apartment and on to the street isn't going to suddenly mean that Romanian cleaners won't be living 7 to a house in zone 5.
This housing crisis was deliberately engineered in London as a result of the 1980s war between Thatcher and local government. Council housing was privatized as a result (& some of those gains were capitalized by existing residents through right-to-buy).
Councils were starved of tax funding through the UK's comically low council tax rates. This meant that not only was owning property a much better deal but also that councils lacked the funds to increase the supply of housing. Hence prices taking off like a rocket.
There's your supply issue. Rent control would be a drop in the bucket compared to that.