So I guess your solution is what? Maintain the status quo, and let an oligarchy of authoritarian feudal lords in all but name, exploit and live off their largely inherited, certainly unearned, ever-growing dominant positions?
You are focusing too much on the housing thing, but that is a strawman argument.
A just society does not have to be a society where everyone is equal or has equal things, that would be UNjust because people are not the same and don't contribute the same. People who contribute more can still have nicer things.
Here is how it works in practice:
1) inheritance: I. is the foremost instrument of perpetration of privilege: it needs to be entirely abolished. When you die your entire wealth is put in a pool and redistributed, along all the other dead people's, equally among every newborn. Any attempt to work around this is met with extreme social stigma and contempt, and the punishment of having all of it confiscated and living a-la-Diogenes for the rest of your days. This will only need to be enforced for a couple of generation because the system naturally generates a more equal state of affairs in which no-one will have huge wealths. The system should be tuned to create disparity of no more than say, 10x between the richest and the poorest member of society.
2) income tax marginal rate: nobody needs more than 1 milion a year. Something like this: after 200K the marginal rate becomes 80%, after one million it goes to 99%. If you are going to be an asshole that one million is not enough for you, you probably are an asshole for whom ten or a hundred are not enough, dead set on sucking up all of society's wealth for yourself, like the leech you are. Leeches get salt.
3) socially important goods and services: nothing of social importance can be owned by private citizens: water, energy, trains, roads, sanitation, public transport, the entirety of healthcare, education and social security, as well as essential food commodities need to be held by public ownership or no-profits. Same for banking as banking has a social function. You can still privately hold other companies that are non-essential to the wellbeing and dignified living of citizens.
4) land tax. You cannot own land, the same way that you cannot own air, or water. All land is everyone's land, if you occupy it, you pay rent to the community. If the community makes that land valuable because, e.g. they create a lot of business and infrastructure and educate the people who go to work in the City of London, then that land is worth a lot more *because* of the community and you are going to pay a much higher rent *to* the community.
5) sortition: since career politicians are a bunch of greedy incompetent twats, and the way they are funded is nothing else but bribery under another name, which of course only goes to favour the same notorious people, we are going to randomly pick a certain number of citizens to represent us for a term of service. A bit like they used to do with the draft conscription. The position cannot be declined, and comes wtih huge respect and honours and benefits from the community (I guess we know who to give those mansions that you are worried about now)
One term and it's over. No second mandate is possible, even if one was so lucky as to be drafted twice.
Done. Everything else is going to fix itself.