Hi Enerd!
First things first! Regarding the first picture, on the left side there is corrosion as well! Please clean the whole motherboard with isopropyl alcohol.
Regarding the 2nd picture, you already did a lot to clean it. My guess is that a new button sticker will work and I would try this first. If not, you can remove the (new) button sticker and remove the rest of the dirt with a tiny flathead screw driver ***(be REALLY careful with this!!! Do not touch the traces, as they can be cut easily!).*** And after you cleaned it with the flathead, then use isopropyl alcohol again to clean it. Then after this, place (again) the new sticker button on it and test again.
I would definitely buy the new sticker button (2 of them just to be sure) to try. A new motherboard is rare to find and if you can find one, it's probably pretty expensive. Sticker buttons are pretty cheap and worth a try!!
I wish you good luck and let us know how it goes!