It does look at the wrapper. The problem is that whatever decision P makes about Q, Q then goes and does the opposite. Consider:

  def P(program):
    if program halts:
      true
    else:
      false

  def Q(program):
    if P(program):
      false
    else:
      Q(program)

When we call Q(Q), we call P(Q). Let's say P(Q) returns false; P says that Q halts. In that case, the conditional fails, and we then call Q(Q) infinitely. Q has done the opposite of what P said it would do.

Now let's say that P(Q) returns true; P says that Q does not halt. In that case, the conditional succeeds, and Q returns false - it halts. Again, Q has done the opposite of what P said it would do.

Here we have a situation where Q will always do the opposite of what P says it will do. We've constructed a situation where P is always wrong; we've created a paradox. The only way out of it is to say P cannot exist.