Your "liberties" are so broad that you are missing the whole point of the question of P vs. NP. The question of P vs. NP is a very precisely defined mathematical problem. It is not merely a vague, hand-waving question about things that we observe in the universe that we live in.

A summary of the statement P = NP as "anything that can be verified quickly can be solved quickly," or "anything that can be guessed quickly can be solved quickly," can perhaps get the general idea across, but a summary like that is far too vague to actually work with the P vs. NP problem. If your arguments about P vs. NP are based on such vague ideas as "guessing," then you cannot possibly be doing anything meaningful with the problem.

So to find such path, I construct a random sequence guesser and set it loose. Sooner or later it will guess it, right?

If the graph has a Hamiltonian path, and if your random-sequence guesser produces sequences that are truly random and independent, then yes, this method will eventually find a Hamiltonian path with probability 1.

But if the graph doesn't have a Hamiltonian path, how does your method determine that fact?

If your method cannot answer "no," then it does not solve the decision problem.

Surely, it wasnt magic. The environment affected the random number generator in a specific way, which is entirely predictable if we have very very detailed records of how things got to where they were.

Since you appear to want to talk about the actual physical universe, I will point out that this is not necessarily true. To the best of my understanding, the current models in physics imply that certain quantum mechanical behavior really is truly random.

But okay, for the sake of argument let's pretend that we live in a deterministic universe.

What I'm saying in my original post is that, realistically, you cannot treat a TM as "external" or "internal". You can in mathematics, but not in this real world we live in. That means, we have to treat everything that can possibly effect our deterministic TM as part of that TM's encoding, as it obviously effects how that TM works.

Okay, so the password verifier must be part of the Turing machine itself. Is that what you are saying?

So the correct password must be stored in some form as part of the encoding of the Turing machine itself. Is that right?

Then "guessing passwords" is trivial. Just look up the password that your Turing machine apparently already has and return it. You don't have to guess anything at all, because the Turing machine has the correct password right there. It's part of the encoding of the Turing machine itself.

Is that what you are saying?