> High prices attract investment in more housing,

Often planning laws restrict new housing...

> usually denser housing

...and denser housing

> Rent Control prevents the market signals from working.

Rent control is also unfair, if some people (selected how?) are entitled to it and other people not entitled.

I am against rent controls unless it is a right for all. By this I mean that anyone[1] living in a city can go to the local authority and say "I want a rent-controlled flat" and the local authority would be legally entitled to let them one, building new houses where necessary. This would mean the local authority would be required to act on market signals of people wanting rent-controlled housing.

1: with maybe a few qualifications like residence in that city for a year, citizenship, etc.