It sounds to me like the bill itself was just a chance for everyone to vote to show approval (or not) for plans that were already taking place, as reported by CNN June 29 (note there is a  good video at that link where you can see how the statues are in Statuary Hall.)

The bust of Taney belongs to the Capitol collection and it is just going to be swapped out for one of Thurgood Marshall that they also have. 

The full statues displayed there, though, are objects loaned by the states 

Every state sends two statues of prominent residents to the Capitol as part of the National Statuary Hall Collection, and several states have already made plans to swap out their statues.

so they just will take those back and supply another statue. I don't know the full process, must go through a curator. So now the curator will turn down any Confederates and Confederate sympathizers and say "send something else". So actually no big deal except helps the curator who can say no without taking any blame.

Note

The vote was 285 to 120. Sixty-seven Republicans voted with 218 Democrats in support of the bill.

and it was easy for McCarthy to spin as taking out more DEMOCRATS from Statuary Hall as that's what Confederates were 

House Republican leader Kevin McCarthy said Tuesday he'd support the bill, and noted that the controversial statues were of Democrats.

"I think the bill should go further: Maybe it's time the Democrats changed the name of their party," McCarthy said.

and I would imagine it is also easy for those who didn't vote for the bill to spin it as they believe each state should be able to chose what they want to send.

So you see, kind of a nothingburger, actually, because it wasn't a "permanent" installation anyways, it's just that nobody bothered to change anything, to make any rules at all.