Good points, but I’m not yet convinced.
No, I’m not saying ‘us’ in and of itself is subjective. You are correct in saying that ‘let us go’ is idiomatic. However, I think that to understand this, we need to make a distinction between morphology and semantics. We are agreed that ‘us’ looks like it’s objective. In this particular construction, it functions as a subjective. In a second person imperative, the second person is obviously the subject. Similarly, in the cohortative construction, the first person plural is the subjective (there is no first person singular imperative in English).
As for your examples:
Is this really correct? If I were saying this, I would say, “That man did it, [namely,] he [did it].” The words in brackets represent the words that are understood.
“We’re going to the park, [that is] you and I [are going to the park].”
I’m not going to discount it as evidence, but I’ve never heard anybody say, “I’m happy, me.”
“It’s he.”
This one is an example of the predicate nominative. Whenever we use the verb “to be,” the subject and predicate are both subjective. “I am the man,” “The man is I.” It does sound pretentious, but it’s true.
“Me, I’m going to the park” is a different thing. It’s short for, “As for me, I’m going to the park.” Since I hold that “We’re going to the park, you and I” is correct, this point is not made.
At any rate, we must acknowledge that our language is a dynamic one and it is undergoing changes constantly. What someone one-hundred years ago says is correct might sound ridiculous to us. Similarly, people speaking the same language in different parts of the world say things differently.
Ultimately,
No one uses 100% “correct” grammar, which I why I usually shy away from the “grammar police” stuff. English is a living language, and it’s primary purpose is communication. If the point is made clearly, I have no problem with “Let us go then, you and me.”