I suggest many US Americans would say "I'm going to the stores" even if they knew they were in fact going to a single store.
Where "shop" and "store" are "interchanged" can you explain both where, and when or how?
I see your “no idea what they would have said before” as much more useful than Eytmonline’s "shop" meaning "booth or shed for trade or work" c1300 or "to visit shops for the purpose of examining or purchasing goods" from 1764.
Doesn’t c1300 "booth or shed for trade or work" and mid-14C (ie, c1350) "building or room set aside for sale of merchandise” say all that's needed?
“Store" for “place where goods are kept for sale is first recorded 1721, in American English”, suggests it was never previously recorded in any other variant of English. Is that what you’re saying? Isn’t “first recorded in American English in 1721” very different?
Don’t you think that until modern times - roughly, around the end of the First World War - the useful difference between "shop" and "store" was always that "shop" meant a place where things were made… eg, blacksmith’s/ carpenter’s/ jeweller’s shop? That is now, and was then more obvious in US American than in British English… but see below.
That “The word store is of larger signification than the word shop. It not only comprehends all that is embraced in the word shop, when that word is used to designate a place in which goods or merchandise are sold, but more, a place of deposit, a store house…” before anything else, suggests that English is not your first and might not even be your second language.
"Shop" and "store" do not have have distinct meanings in modern English in general, unless you go to the trouble of averaging out a score or more variants… failing that, the difference is an idiomatic preference based on geography, not anything truly semantic.
“We…” who speak of shops as places in which mechanics pursue their trades, as a carpenter’s/ blacksmith’s/ shoemaker's shop would be more specific than you seem to recognise, even if the appropriate term was “mechanics”.
It might be that simple in US American English; that might also apply to Australian, Canadian, South African and New Zealand, Zimbabwean or any other variant you could name.
Today, the difference is purely geographical: British English almost exclusively uses “shop”; US American, almost always “store”; other varieties make their own way. That almost means British English is
Traditionally, that wasn’t the case. The family tended to live in a couple of rooms above a back-room work-place and a front-room sales space; a back-room shop and a front-room store… if you cared about that distinction, which most people did not.
In British English, the work- and sales-spaces merged into a single “shop” and then the back room dropped out, re-emerging as specifically a "workshop".
In US American English, the work- and sales-spaces remain separate, "shop" referring to the back and “store” to the front room.
This strikes me as another example of how when there's a clear difference, US American is usually more obviously right than British English.