In the expression "Let's get started," started functions as a participial adjective.
an adjective that is a participle in origin and form, such as burned, cutting, engaged.
Consider the phrase
Let's get drunk.
Here, we use an adjective, drunk, to describe how we are going to get. It modifies "us," as we see when we remove the contraction:
Let us get drunk.
"Start" is a verb in this context, but "started" is being used as an adjective to describe "us." So we call it a participial adjective.
I would not describe this as a passive construction. Because Let us get started is an imperative statement, there is an implied subject.
You let us get started
This construction, with the implied subject, is a standard active voice construction. We can see how it would look in passive voice by swapping the subject and object:
We are allowed to get started by you.
But as it stands, the sentence is in an active voice, and started functions as a participle, or participial adjective.
EDIT: As per the OP's request, we can ignore "Let's" in order to focus on analyzing "get started." The question suggests comparing "get started" with "start." While these do seem to be interchangeable, and for the most part are with the word "started," these constructions aren't always interchangeable.
To avoid "Let's," we'll replace it with a simple pronoun, "They," and a new participle, "exhausted."
They get exhausted.
Here, we can't use any alternative structure, because "They exhaust" doesn't communicate the same concept as "they get exhausted." If there was a particular cause for their exhaustion, we could say rewrite the phrase:
They get exhausted by the running.
Now we could say that this is in passive voice, because it could be written
The running exhausted them.
But suppose a writer wants to express that "they" entered a state of being exhausted without specifying a cause? The writer has few choices but to write "They get exhausted," even if it could arguably be passive or active.
But there is a difference in "They get started," because They are taking the action of getting started; It is not being instigated upon them by some other, implied subject. This is why I consider it to be active voice.