> if the market rate is $1k/mo
(I'm ignoring racial and otherwise illegal practices in this post, which are also a serious problem. Redlining and related issues are huge problems that need to be addressed if we want to have healthy cities.)
The problems (gentrification) start when the market rate is actually something like $2k/mo or even higher due to popularity and limited space.
It would probably be fine if that rate was due to simple inflation, as the ability to afford rents at these higher rates would have similarly increased. This could even be an acceptable local effect if local wages kept up with local cost of living.
However, in popular big cities, wages have not kept up, so if rents are allowed to rise to the market rate everywhere in the city, the entire poorer end of the population - who are necessary for any city to function properly - tend to end up displaced.
In theory this could probably be addressed by proper zoning. Allowing some high-end expensive housing is fine; it just needs to be balanced with an appropriate[1] amount zoning for lower-income housing. I fear this could be another area that is easily abused for both social class or racial purposes, but that's probably a problem regardless.
Other areas have addressed these problems with very strong rent control laws.
As cities grow and the population become more dense, these issues will become more common. The answer obviously isn't a pure free market, though I doubt we have invented the best way to regulate these problems. Maybe someone here can direct some innovation to the topic of local government and finding a better way to guarantee the wide variety of people that are needed to run a city are actually able to live in that same city.
[1] I have no idea what amount is "appropriate".