Dumbledore didn't order the map to obey on his authority as Headmaster, he summoned a tool with which to fiddle with it. How do you know that crystal rods aren't a standard way to affect enchanted objects, and the one kept in the Sorting Hat is just unusually high-quality?

I mean, if it was created by the founders, the Marauders would have had to do something very similar in the first place. So either Dumbledore undid the Marauders' alterations, or he altered their (or whoever's) creation the same way they would have had to, counterfactually. How do you distinguish between the options?

The Map was an extraordinarily powerful artifact, capable of tracking every sentient being on the school grounds, in real time, by name. Almost certainly, it had been created during the original raising of Hogwarts. It was not good that errors were starting to pop up. Chances were that no one except Dumbledore could fix it if it was broken.

And the Weasley twins weren't about to turn the Map over to Dumbledore. It would have been an unforgivable insult to the Marauders - the four unknowns who'd managed to steal part of the Hogwarts security system, something probably forged by Salazar Slytherin himself, and twist it into a tool for student pranking.