The connection with the stopped watch is that in both cases there's a solution that works only if you ask the question for which it (the watch, or the simple solution) produces the correct answer.  It's like using Pythagoras' theorem to determine the length of the third side of a 6-8-X triangle without saying anything about whether the triangle is a right triangle.  Yes, X is 10 if the triangle is a right triangle, but Pythagoras' theorem only applies if the triangle is a right triangle, so unless it's given that the triangle is a right triangle or you can somehow show the triangle must be a right triangle based on what's given, using Pythagoras' theorem is unjustified.  Similarly, showing that the unconditional probability of winning by switching is 2/3 for the MHP (which is what the simple solutions do) when asked about the conditional probability is not a complete answer unless you also say something about why these two probabilities must be the same. 

 :::::::::You don't need to take my word for this - it's exactly what Morgan et al., and Gillman, and Lucas, Rosenhouse, and Schepler, and Rosenthal, and Grinstead and Snell, and the others I keep mentioning are saying.   And, since Wikipedia doesn't care what you think or what I think, and there are plenty of reliable sources that present unconditional solutions without justifying their answer must be the same as the conditional probability it's perfectly fine to present these sorts of solutions.  What I'm saying is NOT fine is presenting these solutions as if they are the undisputed way the problem should be approached.  This isn't a difficult concept - I'm perplexed why we have to keep arguing about it.