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.
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.