Unless I am missing or overlooking something in the bill's text, it looks to me like it's just a bill to stop nationwide injunctions, making it so judge rulings apply only directly to the parties themselves. A desire to stop nationwide injunctions is hardly "Gleichschaltung", and indeed nationwide injunctions were really not a thing until the 1960's, and even then didn't start to become common until even later than that.
To put into perspective the problem of nationwide injunctions: Suppose a new federal policy is implemented, and due to it being controversial, a lot of people in different areas sue over it. The question goes in front of, say, 45 different district judges. 44 say the policy is lawful and constitutional. The 45th says it isn't lawful and issues a nationwide injunction. Congratulations, that
one single judge has set judicial policy for the entire country (at least until appeal, which can take a while), even though they were only 1 out of 45! And with the power of forum shopping, a plaintiff can dramatically increase their odds of making sure their lawsuit gets heard by a judge likely to be that 1 of 45!
In fact, the arguments against nationwide injunction have been strongly articulated by outlets that are no fans of Trump:
See how GOP Senate obstruction lets Texas courts block Obama federal rules—from immigration to LGBTQ rights. Discover what’s at stake!
www.dailykos.com
Sometimes, it’s better to give your opponent what they want.
www.vox.com
Nationwide injunctions have become a serious problem in American law.
www.theatlantic.com
Daily Kos, The Atlantic, and Vox are not conservative sources. And they were all complaining about the practice of nationwide injunctions.
Though there's the rub: Such articles by liberals appear to have not been common when Trump was in power, but were more popular when Obama or Biden was. Thus, liberals were complaining about the practice of nationwide injunctions when it was used by conservatives against them... but largely seem to have clammed up about it after it was being used by liberals against Trump (the linked Vox article, written while Trump was president, is an exception)
Of course, this inconsistency applies just as much to Republicans. The things this bill would stop? Were exactly the things conservatives were perfectly happy to do while Biden was in charge. But now that Republicans are in charge and it's
liberals who are using nationwide injunctions against them, all of a sudden this is an issue that needs to be stopped. If an identical bill to this was introduced 1 year ago, I expect many (perhaps even most?) of the people voting for it would have voted against it.
Still, however inconsistent the parties' stances may have been, a desire to halt nationwide injunctions is absolutely something that people bitterly opposed to Trump have argued for.