Ofc construction would continue just as it did for thousands of years - and because of the exact same reasons rents can get that high: people just need to live somewhere. Homes have intrinsic value and are necessary regardless of monetary value.
It's not like the housing market didn't exist before the value increase in the western world (so in the last 70 years).
There would however be less investments into homes, less financial incentive to build extra homes you don't intend to use personally. So yes, fewer places to rent, more homeowners. And secondary market would function just as good, people move around all the time, needs change over ones life, etc - buying and selling would be easier. Prices would still vary a lot, but would reflect the majority (more democratic prices?).