No fan spin until overheat?

I bought an Xbox 360 E brand new a few years ago; it worked fine until a couple months ago. Now, when I turn it on, regardless of orientation or ventalation, the fan won’t spin even when coaxed with a can of air. If I launch a game it overheats after a few minutes, does the whole overheating process where the fan does spin at max speed, before shutting down. If I reboot immediately, the fan turns on right away at a normal speed and the console never overheats, until I turn it off for a day and come back and start the whole process over. I’ve replaced the thermal paste with Arctic Alumina but nothing changed. What’s even weirder is if I don’t touch it after boot, leaving it at the profile login screen, the fan will eventually start spinning, beginning at max speed and then going down to its normal quiet operating state. Should I just modify it so that it’s hardwired to 12v / 5v? Or is there a more simple solution?