Usually, when playing "core only", I don't allow PrCs, or only allow those that exist to plug holes in the multiclassing system (Mystic Theurge, Eldrith Knight, Arcane Trickster). To be fair, I also tend to use a very small number of house rules (fixed hp/level, a couple of tweaks to multiclassing, that sort of thing).
Does this make my game not-core? I don't care.
When I'm running a game, it is my preference to play core-only, or to use only a small number of carefully chosen supplements. When playing, I'll go with whatever the group uses, and not complain either way. Certainly, the full spectrum of games can be enjoyable, dependent on the DM, the players, and the circumstances of the campaign. (It's also worth noting that if I were DM, and my group as a whole came to me and said they really wanted to play with supplements allowed, I would be willing to compromise on that.)
One thing I do object to is when people choose to look down on those running core-only (or supplement-light) games as being somehow inferior. I expend a fair amount of effort running the best game I possibly can, and am constantly looking for ways to improve my game. The suggestion that I'm somehow shorting my players simply because I choose to exclude the vast majority of the published material isn't really fair, IMO.