Some will sell, yes, but not all. Others will just sit on property and wait for appreciation, or use it as a second home, or keep it empty just in case they end up needing it again or for something else.
And rent control will increase the attractiveness of renting compared to buying so some people who would have wanted to buy will rent instead, and they are likely to be more financially well off, which will make them more attractive renters than the people who don’t have the financial capacity to buy. So, again, the worst off will be poorly served by the policy.
This isn’t just idle academic theory. Governments all over the world bring in rent controls again and again because voters like the sound of it. And every time rent controls are brought in it restricts housing supply unless a lot of other regulations (which have their own unintended, negative consequences) are also brought in. And it still does nothing to tackle the real issue which is too little housing.