Did this mechanic actually look at the car? Or was this just a guess over the phone? If no one has inspected the car then you don't really even have a "first" opinion, so you can't really get a second here.
Here are your most likely basic options if the wipers stop working: 1) fuse; 2) bad connection at motor plug; 3) motor died; 4) linkage came undone.
It sounds like the mechanic guessed # 4, but that is unlikely to produce any burning smell. #2 & 3 would most likely give some smell. As for #1 - the fuse is supposed to burn out if there's a problem that might involve any burning.
Anyway - you must get this sorted out as you really don't want an electrical fire or a possible simple electrical problem turning into a much bigger one. Someone needs to actually inspect the wiper system. These are not that complicated, and the only part that might get a little expensive would be if the motor burned out.