I can't recall ever seeing a switch "back in the day" that didn't have ASICs. There are a few newer switches that do, in fact, use CPU rather than ASICs, but they're built using MUCH faster CPUs than what are included in most -- if not all -- routers.
Early days, the ASICs didn't even exist yet. It wasn't a choice, it was all that there was to use software. Then the ASICs came along and obviously dominated. But yes, it always requires throwing more hardware at it.
How far back are we going here? My first experience with a switch was with Alantec/FORE ATMs (circa early 91/2). These had a RISC SCP processor to run the OS/management functions and custom ASICs for switching. After that, in the more mainstream, came Kalpana and then Cisco. To the best of my knowledge, there were no switches available prior to thicknet (first commercial ethernet).
I'd love more information if you have it available. I love the history of tech.