Let literally means "allow." In many cases where this is said, the person saying it isn't blocking/preventing/gatewaying anyone from going. So the literal meaning of "allow" is not intended.
Let also means "to cause to" as in "let me know", or can be "used in the imperative to introduce a request or proposal", as in "let us pray". (Or "let there be light.")
The definition you're referring to matches definition 2a, "to give opportunity to or fail to prevent", or definition 4: "to permit to enter, pass, or leave".
"Let's go" absolutely means "let us go". There's no way around it. It's just not the version of "let" that you may be used to, but that doesn't change anything.
"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.)