"A-Mods" has only been a term since the rise of B-Mods and Sport Mods, which have started to outnumber the top modified class at many tracks due to affordability. I kinda hate the term. It's like calling the movie Rocky One.
Yes, that is the lineage of the IMCA modified. For a long time those cars (and their UMP brethren) were required to use a "stock" Chevelle front clip, from 1968-1972 I believe. Those rules kept getting bent until they went by the wayside and they became fully tube frame chassis.
What I saw growing up in the Midwest is that late models were usually a track's premier class in the 1980s, but priced themselves out of working as a weekly class at most tracks. IMCA-style modifieds became the go-to premier class for a lot of these tracks in the '90s and '00s, and more money and development went into them. The top drivers were rolling out brand new manufacturer-bought cars every year, just like late models. Then UMP and USMTS started touring with them, paying $2-$3k to win, and soon enough these things became late models on skinny tires.
Today it costs so much to be competitive in an "A-Mod" that at a typical track, you might see 15 "A-Mods" in the pits and 35 "B-Mods". The B-Mods are usually just a 2-barrel carb version of the A-Mod, often someone's used A-Mod from two years ago.