The market will and does fail in the face of bigotry and discrimination; we've seen it from redlining to, well, now. Addressing market failures is the government's job.
Social stigma might blunt these effects but it's unlikely to eliminate them any time soon. I bet a lot of the people who (effectively) refuse to rent to black people don't think they're the slightest bit biased against black people. They're not keeping track of their individual decisions about rental applicants, and they can easily let their brains fill in the blanks to tell them there were all sorts of negative signals they picked up on when declining reservations.
You can be optimistic all you like that we'll "eventually get there" (I'm not) but optimism doesn't do much for people whose rights are being infringed today.
Because that's what it sounds like you are saying.
And, historically and into the present, there are cultural incentives (racist-ass racism) and economic incentives (risk aversion) to make that happen.
The next step is that the discriminating companies are out-competed and go out of business, leaving only non-discriminating companies. Conclusion: that's why there are no discriminating companies left!
This is a nice thought, but it barely works in theory and I've never seen it work in practice. That the consequent isn't observed regularly should tell you something about the assumptions we started with.
Depending on how bad the problem is you could end up with ghettos and other nasty things because of people who aren't able to get any sort of service.
Why? If someone chooses to discriminate, why not let them suffer the consequences of lost potential revenue naturally?
It gives people who don't discriminate a chance to step in and gain market share.
It reminds me of the story of the bakery who wouldn't make a wedding cake for a gay couple. If I owned a bakery, I would put up a sign the next day saying "we will happily make cakes for gay couples".