I think the "parties are private organizations" argument would be more convincing if in practice they didn't have such a large influence of the regular public election process.
With something like a ranked-choice voting system, and less influence of party politics on public elections, closed primaries would be fine. But if we're gonna have two heavyweight private parties that exert a ton of influence over public elections, then I think open primaries should be the norm.
I think right now, it's kind of the worst of both worlds. Obviously, it's not "disenfranchisement" in the literal sense of "not being able to vote in a general election", so I'm not necessarily saying it's equivalent to other voting rights issues, but in practice, it is another hurdle that makes it more difficult for people to participate in democracy. And ideally, we'd be removing as many barriers as possible, not adding more and using "well, technically..." arguments.