Once PS3 and Xbox 360 came in picture, things changed slightly since they relied more heavily on their network services… albeit, PSN back then was free for online gameplay. These were on the same generation as Wii, but Wii was a winner for casual playing.
You've got the option of OG authenticity with CRT emulation (although I use a shader emulating a high-end analog RGB CRT so the effect is more subtle) or you can 'upgrade' games to looking and feeling quite modern yet without the modern cruft. Also want to give a nod to PS3 / X360 emulation as both have come a long way in the last ~18 months and I'm really enjoying working my way through a backlog of titles I never played back in the day.
Mind you, I broke my video game addiction shortly after that time-period - my last console was an Xbox 360 - and haven't got a firm grasp on what's come since.