"with people having apparently similar but in reality completely unrelated problems" Not sure what he was reading - this is a known problem with at least many early 2011 MBP - I had one (still do - it's in the closet). Apple is refusing to acknowledge this as a problem, and will offer to replace the motherboard (or something like that) for ... $300 and approximately 5 business days. From what I've read, whatever they…
I have an early 2011 MBP and last year the same thing as OP happened. Ended up bringing it to Apple and did their repair depot option for $300. At least they extend the warranty for 90 days.
Not looking forward to this happening again, though. Especially since last time it came back with a couple of scratches on the bottom case that weren't there when it was sent out. I guess we'll see.