I still have my PS2 but I can count the times I used it after getting a PS3. Maybe 2 or 3 times. So my answer would be no, no need to have all 3 plugged because I hardly play old games at all. At best, keep the older consoles boxed and unbox once you have nostalgia feelings. Then box them again for a few more years.
I think backwards compatibility is overrated imho. I buy new consoles to play new games. The old games I'm most likely sick of/tired to play anyway. Gotta move on, just watch some youtube video and the will to play goes away. Also, many times I come back to an old game I realise how bad it is compared to new games - bad graphics, etc, or how badly it aged and put it aside pretty quickly...