"Let's go" never means "let us go". Just try to articulate it as such! I can't.

"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.

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toast0 3 hours ago | parent | prev [-]

Shall we go?

Let us go / Let's go / Let's

If you don't want to use the full form, it shan't stop me.

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esafak 3 hours ago | parent [-]

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?"

	▲	ninkendo 2 hours ago \| parent \| next [-]
		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.)
	▲	efdee 2 hours ago \| parent \| prev [-]
		You can't always replace "let us" with "let's", but you can always replace "let's" with "let us".