VCR started to change this; you could now record that episode when you had something else to do; in 1985 about a third of people owned a VCR, by 1990 about two thirds. Later DVD and streaming changed this even more; I remember reading about a bloke who had all of TNG on videotape; it occupied an entire wall and must have cost a small fortune: 89 tapes (2 episodes per tape) it must of cost about a thousand, give or take. With DVD you can but all seven seasons for an affordable price and your girlfriend won't hate you for taking up a wall, and with streaming it's even easier.
Star Trek has a somewhat interesting broadcast history; from 1966 to 1969 it was broadcast in the traditional way: a set timeslot on a particular day and by season. It had a dedicated set of fans during the original run, but was never a phenomenon like it is today. After the series ended some regional broadcasters got the rights and started airing it ever day during meal time. There's about two months worth of material, and after that it would just start over again. It's that format that made Star Trek truly popular, eventually leading to the films, TNG, and all the rest of it. Today it's a common way to air older shows.
In 1983 Babylon 5 would have been a non-starter, simply because few people would be able to watch all of it, and you do need to actually watch at least a significant portion of it. Even with multiple repeats it takes a lot of effort to consistently follow a show with traditional TV. I never watched Babylon 5 when it originally aired; I watched some episodes but I had missed many and I was generally just confused by it; it wasn't until some time in the early 2000s when I downloaded and watched it that I actually followed it.
I think there are a few shows in the 90s leading up to the Sopranos: Twin Peaks (1990-1991), Homicide: Life on the Street (1993), Babylon 5 (1993), Oz (1997), and The Sopranos (1999). Maybe some others I'm not familiar with.