Why playing is important...
If you were designing the interior seating and controls of a car would you start by never sitting in one or driving? You might be able to eventually design that interior, but you would been reinventing wheels all over the place and you would likely build an interface that would alienate users who were used to the steering wheel + 2 or 3 pedal conventions.
Understanding genre conventions:
You should always take a look at the top games in whatever genre you plan on making. Your players are going to attempt to interact with your game based on expectations of having played those other titles. That doesn't mean that you can't deviate from the established conventions of a genre, but when you do deviate it should only be for something your title is focusing on.
See how things go wrong:
I would also suggest playing a few of the lowest rated games in your target genre. Learning from good games is often hard as it can be difficult to nail down precisely what makes a system work since it can be effected by multiple elements. On the other hand with bad games, it is often incredibly easy to get a feel for what doesn't work. This lets you avoid making those same mistakes in your game.
Do designers play? Yes, all the time, and not just video games. Boardgames, card games, miniature games, physical sports. Playing a wider variety of titles is a great way to pick up on little mechanics or rule combinations that you can add to your toolbox for future titles. However, there is a big difference between playing a massive number of games, and playing a massive amount of one game. Spending massive numbers of hours in WoW or Call of Duty or whatever your interest might be is fun, but not directly helpful to designing. You only need to spend enough time with a game to see it's range of content and then move on. By all means keep playing for enjoyment, but at that point it's no longer research.