The difference between the player and the designer is just that the designer has put time into thinking about design, and with that time has probably acquired a broader experience and more developed conceptual framework of what a game is and how it functions. With that experience and theoretical foundation the designer has mental tools to test ideas (and how they fit with other ideas) that an ordinary player doesn't have, and the designer also has the habit and mental toolbox to produce more ideas faster.
Very well said.
Every designer has to _be_ a player, just like every writer has to be a reader, every musician has to be a music fan, etc. If you're not consuming it you could make random attempts at producing it but you're not going to have any feel for what's functional and enjoyable.
One possible counter-example to consider would be Shigeru Miyamoto. Despite arguably being the world's most famous and successful game designer, he reportedly spends very little of his time playing games. I'm sure he plays the games he's working on, so your argument may still hold in that regard, though it would seem that one can still theoretically be an incredibly successful designer without actually gaming regularly.
(But don't get me wrong: I'd certainly agree that playing others' games would have a much greater benefit to a designer than not doing so.)
