Bought the PS2 day one, the only PS1 game I played with it was FFIX, which was released after the launch.
Bought the PS3 with BC in November 2007, and it died in August 2012. The only PS1/2 games I've replayed with it have been RE2 and SH3. Well, I would have played the first SSX also, but it didn't work properly on the EU launch unit.
So, for me personally BC isn't mandatory. Of course it would be more convenient to be able to play the PS3 games on PS4, as I don't really have space to keep two consoles on my small shelf. But looking at the past, I don't tend to play the games of the previous generation that much after the next one has launched.