Xbox 360 E shutting down due to "insufficient ventilation", fan doesnt work/ underperforms

My Xbox 360 E is shutting down due to "insufficient ventilation" after a couple of minutes being on, even though both the console and the power supply are in well-ventilated spots. Fan doesn't work / under-performs. My console doesn't seem to be able to
work for more than 2-3 minutes. After that 2-3 minute period an error message pops up saying that the console will be shut down due to "insufficient ventilation". A few seconds later, the console turns off and a red light starts flashing, sometimes surrounded
by a static green ring and some other times without that ring (i don't know if there are any differences between the two occasions, the point is the flashing red light). Even after restarting the console multiple times, clearing cache, resetting the console
to factory settings, removing all the storage spaces other than the memory unit (I only use a 250GB Hard Drive), and pretty much anything else that could be done to "maybe" fix the problem, nothing seems to change. Because of that i have 2 questions for that
matter:

1. If i connect a USB intercooler/cooler to my Xbox (for example the Nyko Intercooler STS for Slim), will that make my console fan start spinning or should i just consider purchasing a replacement fan?

2. If I purchase a replacement fan, will i be eligible to replace it manually by myself or should i just consider sending it to a repair center so they can do it? (Notice that i want to spent the least amount of money possible, and replacing it by myself
will most likely save me more money)

Hopefully you can give me some insight to my problem. Apologies for any grammar/vocabulary mistakes, and Happy Holidays!

:)