...once you admit that there can be restrictions on property rights, it's unclear how property rights in and of themselves is a steelman of private equity. Why not just say that "you can't sell to private equity" is one of the restrictions?
What do you think that would look like in practice? Private equity is investment in non-public companies -- the purchase of shares in non-public companies or the purchase of the companies outright. In many cases, the private equity firms themselves are public; but in many cases, they are private, as well.
There isn't a way to make a rule that "you can't sell to private equity" but there can be a rule like "you can't sell a private company at all" or "public investment funds can not purchase shares in private companies" -- if you try a few of the variations that are possible, you'll see that they're all bad rules, I think.
You really have to think about hard about what you mean by "a private equity company". It's not as easy as it looks. If a manufacturing firm starts to buy up suppliers (many of which are small, private entities), is it a private equity company? It's buying up non-public companies.