Exit makes sense as a human right. This is true across many levels of organization: you are free to quit your job, drop out of school, even emancipate yourself from your family after a certain age, and it's widely recognized that protecting the freedom to do these things improves collective outcomes.
A corollary is that you should be allowed to join organizations that want you.
But entry to any organization, regardless of whether they want you or not? In a world where Google has to hire everyone who applies, or Stanford has to admit everyone, or families have to house and feed anyone who shows up at their doorstep even when they don't want to, you no longer have a Google or a Stanford or a family functioning anywhere close to its current level. "Entry to anywhere as a human right" is an idea which may be initially attractive due to its simplicity, but the only people I know who stick with it after further thought appear more interested in harming groups of people they don't like than trying to lift up everyone.
And as a practical matter, for all of its flaws, the US has nevertheless done more than any other country over the last 7 decades to enforce a right of exit across most of the globe. A more diverse and stronger US would be an improvement, yes, and I'm pretty sure Canada/Australia-style immigration policy will get us there. But there are ways to become more diverse and weaker, too; and importantly, in that world, bigotry is even further from being solved, because the probable leading superpower is a more bigoted, monoracial China. (In contrast, when the City College of New York ceased to be a top-tier institution, the damage was fortunately limited to just itself; other top universities had ended anti-Jewish admissions practices some years earlier.) You seem completely blind to this danger.