Interesting question. My initial reaction was as yours: That it started out as ‘let us say’. French uses the expression ‘disons’ in exactly the same way: to indicate something given as an illustration or for the sake of argument. French has a first person plural imperative, where English has to cobble one together.
OED recognises this parenthetical usage of ‘,say,’ as being equivalent to “,shall we say?” or “,let us say,”. So you have that support, though I could not find a specific example of its use from OED and so could not trace its origin.
I could find nothing helpful in the Oxford English Grammar either. So I am left with the question: can this usage be parsed? I don’t think it can. It might stand for ‘shall we say?’, which would make it first person plural future active indicative interrogative; It could be first person plural imperative active.
I would suggest that it is what in ancient Greek I would call a signpost expression: a signal. There may be a technical term for it, but I am not aware of it.
What ‘,say,’ is doing is to send an instruction to our brains about how to read or understand the next word(s) or even the sentence as a whole. It is saying something like “don’t take this literally as the only situation I want to know about”, or “this is for illustrative purposes only”. So If the insurance agent of your example gave an unqualified ‘yes’, knowing that if you took your car off road and got stuck or or had it stolen, having left your keys in the ignition the answer would be ‘no’, she would not have responded fairly.
There are other such verbal signposts: as it were, to apologise for a somewhat metaphorical way of saying something, could just about be parsed but that is not the point; bless him is not a request to the Almighty to bring blessings on the small child that has just done or said something appealingly funny, parsed as imperative as it may be. They are, in the sense I have described, ‘signpost expressions’: parsing, even where possible, is not the point.