Rent control still leaves people who want apartments without them, it simply changes the allocation strategy from "willing to pay market price" to "was here first" or, in the case of designated affordable housing, lottery.

Increasing density limits, or improving transportation to allow wider sprawl, actually add housing supply in a way that might keep up with demand.

Montreal is affordable, but I don't think you can attribute that entirely to rent control. Forty years ago it was the premier city in Canada, but it's had a serious decline since then. Perhaps not an absolute decline in population, but certainly slower growth than the rest of the country, and way less money than it used to have.

I'm no expert on the Montreal rental market and rent control rules but if you are not experiencing shortages then the rent controls are likely not "binding". That is, the controlled rental rate is very close or above what the rental rate would be without rent control.

Since it appears that Montreal has relatively low rental rates it is likely that supply is high but demand is low relative to other cities. This is consistent with the relatively low population growth of the area (the Island of Montreal had a larger population 60 years ago than today!). In this situation rent control would have very little impact and be unnecessary anyway.

Rent control is good for people who already live somewhere, and bad for new arrivals. Places like the Bay Area, with huge numbers of new arrivals and not much new development, see this more than "normal" cities.

The issue being that renting in the city effectively becomes a lottery, because not everyone can live there that wants to.

Montreal city has today a lower population than it had 50 years ago (using current city limits). Abbreviated from Wikipedia:

  1966	1,750,969	
  1976	1,664,527	−5.7%
  1986	1,541,251	−0.9%
  1996	1,550,369	−0.2%
  2006	1,620,639	+2.3%
  2014  1,731,245	+5.0%

So I don't think you can attribute the affordability to rent controls at all.

You can perhaps attribute it to the rent controls that all the metropolitan population growth has gone outside the city (where it has over the past 50 years doubled from 1.3 million to 2.6 million).