> In effect, those willing to pay the most are replaced by a random subset of those willing to pay PRICE_CAP or more.

This is precisely the effect. And many would consider this fairer. I think it depends on whether you people's willingness to pay as being determined more by their preferences (between say convenient transport or having a bigger house), or their income level. If it's the latter, then I think it's fair to question for each good and service, why should someone with a greater income be prioritized?