Hmm. Government continuing uninterrupted, okay; but appointments and acts can be done by conventional means could take years and years and be contested by those who are fanbois and adherents of the one who committed criminal fraud to be president. Plus some of the fanbois and adherents were in on the criminal fraud.
I don’t think there is any precedent for this particular situtaion.
You are right. There will be no instant gratification.
For example, appointments with the consent of the Senate will remain valid because it is presumed the Senate would have approved the appointee regardless of who nominated them.
No court is going to declare routine actions signed by a Treasury Secretary invalid for such a reason.
And we already have a president who can undo any executive orders of a previous president whether or not legitimate.