Have.. have you seen more than the clips? The time is ambiguous in the movies, the first Mad Max is set in a crumbling society. Road Warrior, Fury Road, and Beyond Thunderdome are all well after relevant governments have collapsed.
Beyond that... it's not that they don't CARE about the cars it's that you CAN'T do the proper maintenance. You can't wash a car without a ready supply of water. You can't do an oil change without oil. Hell, the plot of the second movie was about the villains trying to procure fuel.
In the first movie, he was driving his police issued vehicle and then the special vehicle made for him before the police department collapsed. So in that, he never stole a vehicle, when there is no owner, it can't be stolen. And, looking it to be sure, that same vehicle was used in both Road Warrior and Fury Road, both times maintained as best he could keep it. He uses something else in the third movie, but I couldn't be bothered. I think we're all trying to get beyond Thunderdome. A-ha.
Anyway, aside from his vehicles, Mad Max either uses things offered to him or hops on those of people actively trying to harm him, which, you know. No one should object to.
And finally, the cars aren't a "mulligan" and I'm not sure if you are using it incredibly poorly. Since you are comparing it to Rosebud, it more makes me think you don't know what a "MacGuffin" is. Either way.. neither works. The Mad Max movies really don't have much in the way of MacGuffins. Only the two Road movies come close, but neither really work. As the fuel is actually used in the plot of Road Warrior it's not simply a thing to start it, I'd be generous and not count it. In Fury Road the pregnant women are actual characters, so again, I'd not count it. And trying to compare it to a "mulligan" also fails, as no vehicle is just randomly discarded. If a car is obliterated in a chase, yeah, they can't use it, but the groups try to keep an maintain the vehicles they use, the ending often having them riding them into the sunset with what they finish the finale in.
Every point you raise fails. Maybe you should have watched the movies instead of just a few clips?