Each State has a vested interest in seeing that the rules applied to them are also applied to every other State. That makes Texas a proper plaintiff.
No, it doesn't. And the Supreme Court agreed. Even the two justices who believed the Bill of Complaint should have been accepted would not have granted the requested relief. The Bill of Complaint issue is related to the standing issue in the sense that it was the reason the Bill was rejected. However, you could argue (as I believe Alito and Thomas implicitly did) that the Bill of Complaint should have been permitted, but the case then dismissed due to lack of standing.
I'd just point out that by your logic, other states should have been able to file lawsuits in federal court challenging the Bush v. Gore case in 1992. And there really isn't even a reason it should be limited to states rather than to the citizens of all those states, because everyone has an interest in seeing laws enforced.
But that general interest in seeing laws enforced is never sufficient for standing.