So someone is arrested for flag burning… and the case is immediately dismissed by a judge.
Ok, and? What makes you think they then get released from prison? Cause they just have to?
Pretty much everything the government does to “enforce” their will can be challenged
If the executive doesn't empower the judiciary, then no, everything cannot be challenged. You can assert your right to challenge an arrest before a judge however much you want, if your jailer says "nah", you don't get out of jail. The court is the veneer of slow and steady legitimacy validating the force of the executive, but rulings are just words, the thing that actually makes you have to listen to them is force.
This is, to be sure, a constitutional crisis likely to end with chaos. The real result would likely entail some sort of attempt to stand up a new "Supremer" court or decide that certain types of cases cannot be appealed or governors relying on their state courts and refusing to bend to a Supreme Court ruling they and their constituents don't agree with.