In many cities, new buildings are exempt from rent control if they are "luxury" buildings. So the supply of luxury housing increases. Meanwhile, the incentive to build non-luxury buildings is greatly diminished b/c those buildings fall under mandatory rent control. Also, there is a bigger incentive to turn buildings into condos, etc., to avoid having to deal with rent control. Not sure where you heard that new properties are exempt in most cases. This may be due to the fact that most are luxury buildings.
So there is a finite amount of rent-controlled housing, and since it's such a great deal, the landlords have little incentive to improve the fixtures, furnishings, etc. They will rent the place without improvements, so there is no reason to spend the money on them. Thus the places deteriorate over time and begin to become more and more slumlike.
Rent control also encourages over-consumption of housing space. My 3br apt example above explains one example.
Without rent control, supply and demand can function properly. Maybe this means that cheaper housing is located a bit further from the highest rent area of a city. More likely, if there is demand for cheaper housing, someone will figure out a way to build it. The huge number of condemned buildings in major cities is evidence of the resources that have been completely wasted b/c nobody had the incentive to improve the places enough to make them liveable!
If you rent, it's not a permanent home. This has many advantages -- if you get laid off you can easily move with a month's notice. You don't have to worry about long term upkeep, you don't have to worry about hiring a realtor or trying to recoup your investment if the market dips. The other side of the coin is that sometimes prices change to reflect other realities. An increasing price probably means that the area is thriving and (surprise) the space is actually worth more. Once you get into declaring that things should be priced by public officials rather than based on the market price (what they are actually worth to someone else who's willing to pay for them) you are stepping onto a very slippery slope.