If I got Bitcoin out of no where I would look at the transaction to see where it came from. If I got another one I would look again. 
I wouldn't think of looking for a signed message but if the transaction shows it then I would see it.

I wonder why they didn't make Bitcoin or modify it where you have the option of sending the Bitcoin but you don't release the Bitcoin until the other side accepts the Bitcoin. Once it's accepted you then release the Bitcoin.
That would solve so many problems of sending it to the wrong address. 

 Yup, essentially keeping it in escrow.