> The elected branches don’t have a “mother may I” relationship with the courts.
That "mother may I" phrase implies asking for pre-approval for routine stuff, which obviously isn't what's at issue here.
> Injunctions are by definition “extraordinary relief” reserved for the most unusual cases.
The unusual and extraordinary may have become quite common of late, but that doesn't mean it starts being wrong for a lower court to intervene when the president tries to fire an FTC Commissioner in clear violation of precedent about firing FTC Commissioners. A president ignoring the law is definitely sufficiently "extraordinary" to justify temporary injunctions, and the fact that he's doing it a lot only reinforces the need for the checks and balances.
> A hastily issued district court order should be (and is) subject to being overturned by the higher courts just as quickly as it was issued.
Don't be ridiculous. The entire concept of temporary injunctions is built on the fact that many situations need to be slowed down and the potential harm prevented until serious legal questions can be properly considered. The lower courts aren't being "hasty" in the sense that they're issuing these orders on a whim with too little justification, they're being hasty in the sense of acting with commensurate speed to the executive actions of questionable legality that the courts need to keep in check. But the Supreme Court shadow docket decisions do come across as insufficiently thought out.
> The district court doesn’t have to grant discretionary injunctive relief based on precedents that are in doubt,
A lot of these precedents really aren't in doubt, except to the extent that it's clear the current Supreme Court doesn't like them. But until such time as the Supreme Court furnishes the proper decision overturning existing case law (not just hinting at a willingness to do so in the future), it's reasonable for lower courts to continue applying that case law when it is extremely obviously applicable.