Without seeing the motherboard nobody can really say. It could be a simple thing, or it could be something more involved. I don't think we have a decent source of 101 motherboards either just in case you thought about just swapping it.
It does not sound like a common failure method, though I would ask what game/flash cart you are trying -- shutting off and freezing can be caused by them. It could also be a bad battery, and the ones out there these days are all way past their time.
The cleaning approach could be done by anybody that does board repair -- if you can replace a surface mount resistor on something then you can probably do this.
Buttons not working is usually the underlying pad/carbon traces are no good, but in conjunction with freezing/powering off
As far as specialist repair then there tends not to be any other than those working on specific consoles, and nobody really does 101 stuff (some might do things to install 101 screens on earlier models but that is a different matter). Those doing component level repair on phones, tablets, laptops and whatnot should be able to do this, however we don't really have nice diagrams (
https://circuit-board.de/forum/index.php/Thread/13913-STRIP-CLUB-PCB-Scans/?pageNo=1 does not have SP of any flavour right now but generally is as good as it gets) and I don't think the gigaleaks gave us any. Those working in phones and whatnot often having nice diagrams to reference against such that they can clean things up, reseat components, replace basic passives and go from there. This might change in the future if prices for such things keep going up (for many years new GBA, though maybe not 101, was next to nothing and people shocked you would want it so repairs and knowledge base for it tended not to be a thing).