"Let us go" does not only mean "you should let us go" but it is also the first person plural imperative implying that we go. Whether you shorten it to "let's go" or not does not change this.
Same as how "let us pray" is frequently used as well.
Abbreviation does change it; it narrows the meaning. "Let's go" never means "you should let us go" and "let's pray" never means "you should let us pray".
Nowhere does anyone claim that "let's go" means "you should let us go". The discussion was whether "let us go" automatically means "you should let us go", which it does not.
I don't know if I'm being clear. Say you and your family were imprisoned. You would never demand to be released by saying "let's go!". Your bemused family might well ask "Where, to the other corner of the cell?"
English contractions are weird in general in that it doesn't always "work" to contract two words. Tom Scott does a good video about this: https://www.youtube.com/watch?v=CkZyZFa5qO0
(Example, "Is this a good idea? Yes, it's!" sounds wrong. But "it's" still means "it is". It would just sound weird to use a contraction in that context.)
Somebody else brought up the example of “let’s go!” versus “let us go” - not the same thing by a long shot.
“Let’s” in English has a distinct meaning from “let us”, and that is to politely and casually (but firmly) suggest a course of action.
I remember touring a Polish salt mine a couple of years ago. The guide was very good, but her English had a few quirks, among them that she seemed to like the phrase “let’s let me to show you …”. It’s wrong, but you can immediately understand that she meant “please let me show you”.
Same as how "let us pray" is frequently used as well.