RE: Dell XPS12 Apparent Motherboard Failure
Essentially everything inside this system is permanently affixed to the mainboard - the processor, the memory, etc. That means just about nothing inside is separately replaceable if it fails.
The diagnosis is likely correct. Parts-people.com is a reliable supplier of Dell components.
If you don't have a backup of what's on the SSD, you can use something like this
to recover the data short of repairing the system.
If you can replace the board, the $300 may make sense. If you have to pay a labor charge, it begins to make less sense - just bear in mind that replacement parts carry only 90-days warranty.