@Lost_N_BIOS - Uh, that’s a pity. But thanks for the suggestion! 
Unfortunately I’m based in Europe, and the cheapest UX31A motherboards I can find on Ebay are running at 105-110 euros (around 120 US dollars). Still cheaper than a new machine, though some refurbished newish X-series Thinkpad (e.g. x250 with fHD screen, i5, 256 GB SSD) would run at just 250-300 euros.
I’ll consider what to do.
One more option would be getting a third-party unofficial motherboard for the Asus from China on Aliexpress (cheapest ones are around 75-90 US dollars), though I’ve had no experience with such parts and have no idea on their actual quality.
@Caigo - Were you on ebay.com? Both listings I looked at, and there was many others around same price, anyway both have shipping worldwide that I clicked into (one does not ship to France)
https://www.ebay.com/itm/151450117760
https://www.ebay.com/itm/254376327896
What does third party unofficial mean? Is that some OEM non-Asus board, or just an Asus board being resold by someone (same as ebay)?
@Lost_N_BIOS - I checked both ebay.com, ebay.de and ebay.it (I’m ordering from Italy). The thing with imported boards is that the shipping expenses are of $50. If I don’t find anything else I may pay them, even if it’s a little weird to pay almost as much as the actual price of the component just for the shipment. 
I found some cheaper ones ($70-90) on Aliexpress, but I think they’re OEM non-Asus boards (I meant this, sorry for my poor phrasing).
Here are two of them for instance:
https://aliexpress.com/item/32862050120.html
https://aliexpress.com/item/32904835118.html
The boards on ali dont usually have any stickers on them,or theyre blacked out with marker pen,so theres no serial,model number,or for some boards the mac address.That can be a problem sometimes as the same board,with a different revision could have the bios for a different model.The bios usually don`t have a serial number or a windows key (win 8) . If you buy one and it works, i would sell the laptop and buy something else.
@LittleSnail - Thank you for your feedback!
At the end I actually decided to get a second hand laptop instead: the uncertainties in buying one of those boards from AliExpress were maybe a little too high (quality of the item, and really long expedition times mainly). I’m currently scanning Ebay for T- and X-series Thinkpads waiting for a good deal.
In the future I may try to recover the Asus as a secondary laptop, or just scavenge it for parts (possibly the display, the keyboard if I find a controller; the SSD has an uncommon standard but may be recoverable thorugh an adapter, and the RAM unluckily is soldered).
Thank you really much for your help!
Keep scanning ebay for a spares repair UX31a laptop,or used motherboard.
Yes, I’m saving the research.