Battery looks ok, reads as normal.
I took the back off this morning and expected to see crap in the fan. All looked good there but I blew some air in it. No relief from that. I then removed the heatsink and reapplied some artic silver. No reilef.
I am letting it run right now and it's blasting and slow as hell. The weird thing is that when the display goes to sleep, the fan quiets down. GPU? This thing may be trash.
I am afraid that it is possible the SOC is done. It could be the gpu is failing and it could be the cpu is failing and since the gpu is integrated if either fails there is no repair to be made. You could see how much it costs to replace the CPU? Or you could see if you could find a used air with a trashed screen and good cpu and try to replace it.
My advice is to let it go. As I said before have Apple at least do a remote diagnostics so you can confirm it is a hardware failure that is major. Then once confirmed I don't see a lot of cost effective options unfortunately.
You got your daughter a great replacement that should serve her for many years and be a battery life champ so she is taken care of and that is the most important thing. Maybe you could use it for a future trade in when you buy another Mac or sell it for parts on eBay with the disclaimer that the cpu/gpu is failing. I am sure the case and screen still have value.
Sometimes when you try everything you can think of you have to give in before you spend way too much time and effort on something that is only going to cause headaches.
I really wish I could say do this and you are good to go but I have seen this with other Intel CPU I the past. Sometimes they just fail. The Gpu or CPU just starts failing after several years-usually it is related to overheating for many years.
In any case I hope someone here can tell you something that will get things going again otherwise it may be wise to cut your losses while you are ahead.