I actually agree with part of your objection: the Necessary Being argument by itself doesn't logically deduce Christianity. It gets you to an ultimate necessary foundation, and additional arguments are needed to determine what that foundation is like.
But calling it a "blank check" goes too far. If an argument successfully establishes a necessary foundation of contingent reality, then not every possible explanation is automatically equally good. You still have to ask which candidate actually explains the relevant facts.
For example, "it's an abstract object," "it's an impersonal necessary principle," "it's a multiverse," and "it's a necessary mind" aren't interchangeable explanations. They have different assumptions and different explanatory consequences. You'd have to argue for why one is preferable rather than simply listing alternatives.
And I agree that Jesus is a historical claim rather than something you can logically derive from the Necessary Being argument. That's not a problem unless someone claimed otherwise. The case can be cumulative: first argue for a necessary foundation, then argue that the foundation is personal, intelligent, moral, etc., and then independently investigate Christianity's historical claims about Jesus.
As for "God could just show up universally," that's an argument about what God would choose to do, not evidence that He doesn't exist. You'd first have to establish that a perfectly good God would necessarily give everyone an equally unmistakable experience of Him. Christianity doesn't make that assumption.
So I don't think the argument is "premises → therefore Christianity, case closed." It's "premises → a certain kind of ultimate reality becomes plausible, then additional evidence has to be examined." That's how a cumulative argument works.