I would be wary of forcing people to sell their house before letting them claim a state pension. Having to move from the house you have always lived in must be very stressful, there are taxes associated with selling it, and the lump sum you get out of it once you've bought the smaller house is probably not enough to provide a significantly higher income than the state pension, even if the house you sold was valued at $2m. If at the end of all of that stress of moving and downsizing, you still only end up with pretty much the same pension income as someone who gets it from the state instead of starting with a $2m house, that sounds like robbery to me. The state can just let them keep living in that $2m home and take its cut with inheritance tax if it wants to.
In the UK, everyone is entitled to the state pension, even if they have millions in savings and their pension pays out £200k/year. I don't think there's anything unfair about that, seeing as they almost certainly paid more than enough National Insurance to cover it. I think it's eminently sensible, and more efficient, to do the adjustment by making the paying-in function a little steeper where it needs to be, rather than an expensive and error-prone means-tested system for the paying-out function.