Let's leave out the Let's, shall we? It just adds another idiom to contend with.
Get started is a normal causative/inchoative use of get.
Since get can be the causative/inchoative of either be or have, we have two possibilities:
- He got tired = He became tired = He came to be tired. (inchoative of be, intransitive)
- I got him tired = I caused him to become tired. (causative of be, transitive)
- He got a new car = He came to have a new car (inchoative of have, intransitive)
- I got him a new car = I caused him to come to have a new car (causative of have, transitive)
In the case of get started, it's both be and have. Since get started means 'come to be in a started state', it works fine with be. But since that state in addition arrives at a state of 'having started', a Perfect construction, there is no contrast. Things often work out like this with idioms.
Oh, as for the original question, no, it's not a Passive; if anything, it's a Perfect, as noted. The confusable get Passive occurs when get means come to be, as in get married, get arrested, get drunk. This works best, as you note, with predicate adjectives derived from past participles of transitive verbs, where the difference between a true participle and a derived adjective is irrelevant. So call it an adjective if you like, but since you can't tell the difference in this example, calling it a Passive invokes a rule that is not obviously involved.
EDIT: There are a number of issues tied up with this question about the Passive.
First, let's dispose of the Imperative. Let's get started can certainly be an order, if uttered by the right person in the right circumstances. Even if not, it's clearly an impositive, with the speaker attempting to impose a course of action on the addressee(s). That's because of Let's, which always functions as an impositive, even if only a suggestion, and always includes both the speaker and the addressee in the invitation, for a joint action. This is one of the very few places where English categorically requires a first person plural inclusive reading of us.
Second, the confusing issue with get started is that both get and start are inchoatives.
That is, they both refer to the beginning of something -- an action or a state.
But in this case, not the same something. Get has wide scope, while start has narrow scope,
so what is to start is some joint task that is unbegun so far (say, making supper). In order to start that task -- in order to get started on that task, or to get going on that task, and eventually to get finished with that task -- it is necessary for someone to do something preparatory, like going into the kitchen, setting the table, going to the bakery, etc.
So get started means to do what is necessary in order to start the project. When one has gotten started (American pple of get), one has passed the boundary into the action or state, so one has started, and if the action is not punctual or the state not ephemeral, then one has entered into the event or the state that has begun.
This is not what I would call Passive. Given the vast ignorance of what 'Passive' means in English among the English-speaking, -learning, and -teaching communities, I hesitate to use the term Passive for anything that does not require an application of the Passive-formation rule to a verb phrase. And this doesn't. It does require a past participle, but that isn't the same as Passive.
One would only get a Passive reading of "get started" if one had used the causative -- therefore transitive -- sense of start, as in start the race. Then the race would have been started (by the unmentioned causers), and that's a Passive for sure. But not normally.