Anecdotal Evidence ahoy: I played and ran BECMI, played 2e (and knew people who played 1e) during the 80s and early 90s, and I never encountered the usage of minis, or any consideration of doing so (I realise that many people did, but it wasn't the norm 'round these parts). It was the non-gamers who assumed that D&D required a board and minis to play it.
Contrary anecdote: I played 1e from 1980 on, and attended conventions, so saw lot's of D&D being played. AD&D was played heavily, 0D&D still saw some play, BECMI hardly at all. Mini's were the norm, lots of them, all lead - they got pretty damn heavy, en masse.

Even then, though, there was a generational split. Kids, like myself, who had entered the hobby as teens with D&D usually had only a few minis, for our characters (if that), and improvised for everything else (dice and pencils saw a lot of re-purposing back in the day). Established gamers had armies of Napoleonics or whatever, and were much more into collecting and painting minis, monster as well as character. I fought a lot of Orcs that looked suspiciously like grenadiers back in the day. (Heck, Grenadier was the company that made the second-best fantasy minis - after Ral Partha, of course.)
The idea of a 'board' was an outsider misperception based on the prevalence of family boardgames in the 50s-70s, though, you're right about that. No battlemasts for the first half of the 80s (I first encountered those playing Champions!). Mostly bare table-tops, but DMs with that wargaming background might break out terrain.
By the 90s, things had changed. The old-school wargamers weren't so much in evidence, and RP, including the elitist 'role not roll' BS, was taking center stage. D&D lagged that trend, as always, but I can see how, if you started with BECMI/RC or 2e, you might have missed the wargaming influence to an extent.
Where Mind's Eye Theatre - I mean, Theater of the Mind - came into vogue was with the rise of Storyteller games, and the role v roll 'debate' (a conflict comparable to the edition war in counter productive rancor and futility, but with all D&D firmly on the "Roll Playing" side of the divide). That's also when we got the bad-rules-make-good-games and less-is-more 'Rules Lite' philosophies, and extreme case of those ideas, Freestyle RP, being tried out.