The issue I believe is caused either by the power supply or cooling fan. I say the power supply because you mentioned that it is 3rd party. Unless you order it from a trusted company like iFixit, Fixez, or GoKrazyWireless, you could be getting a cheap alternative with fake promises and cheap parts used to make it. Similar to a cheap iPhone battery, it would make your phone run worse than it was with the old battery. So make sure the battery you are getting is authentic or OEM. If your battery is OEM, then I believe that it is the cooling fan. Throughout the use of your console, the fan will collect dust, and sometimes get to the point of so much dust that your console will overheat. I would recommend finding a video or another online tutorial of how to clean out dust from your Xbox. I am personally leaning more towards this being the issue because you said YouTube runs fine, but games cause the issue. If you listen, YouTube wont cause the fans to turn on because it doesnt require much system resources. Games like GTA and such use a lot more resources and can cause the fans to go into overdrive, especially if there is a lot of dust, heating up the console and causing it to shut off due to overheating.
Hope this helps!
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