Causative, not passive?
"Let's get started" sounds like an ellipse of "Let's get [sth.] started", for example:
Let's get the session started.
The sentence still looks like a passive construction, but this is deceptive: it is not a passive.
It is NOT "we" that get started, but something, for instance a session, that gets or rather that we get started.
To disprove that we have a case of passive voice at hand, let's get the sentence modified: If we used the verb "be", the sentence would only make sense if we said:
The session was started.
Not:
*We were started. This clearly makes no sense. "To get started" is not a passive voice.
This clearly makes no sense. "To get started" is not a passive voice.
In the same vein, 'let's get drunk' is not a case of passive, but a caseofcase of a causative use of the verb "get". Why? Because 'let's get drunk' actually means:
Let's get us/ourselves drunk.
And to prove this claim, we can change the the indirect object pronoun "us/ourselves" without altering the meaning of the verb in question:
Let's get Joe drunk!