Exercise is extremely important.
You don't have to make fitness an obsession, but exercising should be as normal as eating, sleeping and brushing your teeth. It's what intelligent people do to maintain themselves.
Pretending that your body doesn't need these things is like pretending that you don't need to be concerned with bacteria and viruses. It's not really debatable -- if you take precautions against microorganisms, you'll be healthier. Likewise, if you maintain your brain and body the way it was meant to be maintained, then you'll be healthier and they'll function better.
A lot of people are still working out in an archaic, obsolete way -- like you're stuck in a factory, and it's your job to churn out X number of widgets in an hour. It's incredibly boring and demoralizing to exercise that way ... counting sets and "reps." It's no wonder so few people stick with it. It's a grind.
A far better way is to exercise with variety and short bursts of high-intensity. CrossFit is the most popular brand name for this modern mode of training, but there are tons of others.
My personal routine is to focus on short-burst exercises, with sessions that are no more than about 5-10 minutes of warm-up, followed by only 15-25 minutes of actual exercise (with short 30-second to one-minute rests throughout). Almost all of the exercises I do consist of sprints and bodyweight exercises. The only equipment you need is a pull-up bar, which can be found in a park, which also gets you outside in the fresh air.
The grip strength you develop will improve your playing. Your mood, mental focus, and overall stress-tolerance will, too.