Little Big Planet is another one. The floaty physics sucked then and they suck now, but the interactivity of the levels was above most platformers. The art style felt fresh. Many other games imitated the "crafting supplies as videogame textures" thing. It had 4p couch and online multiplayer. The level editing, the social element of leaving Likes and comments on community levels, even using that stupid Eye camera to take your picture... it all felt like an obvious videogamification of where the Internet was headed. The modding/customization aspect was very PC and this game was influential in bringing it to a console normies.
Then the second game came out and added circuitboards and controllable camera, which players took and broke and made games-within-games. These came out in 2008 and 2011 respectively. Minecraft Survival mode was in 2009, so Little Big Planet definitely gets some credit for popularizing this stuff with the mainstream.
Wii Sports is another favorite series that came from this era. It's fun with all ages in short bursts, never really gets old unless you overplay it and wear it out. The series got better with Wii Play, Wii Sports Resort, and now Switch Sports. The simple toy-like amusement of the motion controls is enough to keep me trying.
yes I realize that i cannot be accepted by D-Pad as a hardcore gamer by admitting these things