You and simonask are speaking at different levels of literality.
Yes, literally, "let's" expands to "let us". But idiomatically, "let's/let us <do this thing>" does not mean "allow us to <do this thing>"; it means "I am requesting that we now <do this thing> together".
Now, I'm not entirely sure why simonask felt this level of literality was a useful one to bring up here, but it is true.
True, but the point was not that they were asking permission, it's the "let us do this together" meaning to which the OP takes offense. He feels like it implies he cannot do it on his own.
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.)
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”.
I dislike the dishonesty. Compare to this line from Office Space: "I'm gonna need you to go ahead and work Saturday". Here go ahead implies that you're being given permission to joyfully do some work you were eager to do. In the Microsoft example, let's implies that this is a bright idea for something fun for you to do with Microsoft, your friend with your best interests at heart.
As a non-English speaker, my understanding of no ahead did not have any joyful connotation. It was rather to express that someone will need to do something that has an initial friction, so not enjoyable.
Your understanding does not match the broadly accepted idiomatic meaning of the expression. The humour comes from the implied inversion of sacrifice, a kind of irony.
I mean, it literally does, but language is not literal.
For the record, I also dislike the familiarity.