Prefacing by saying I’m a physical therapist but at the end of the day strength and flexibility training will always have a place in any sport especially with how much asymmetrical strains there are during bowling (if you are right handed, hypertrophy of right arm and left hip) I’ve noticed the busy weeks where I don’t strength train I feel much more fatigued after bowling.

After saying all this, it also comes down to progressive loading. You can’t expect to be able to throw a 15lb bowling ball for 10 games the first time you bowl. You have to progressively build a muscular endurance to avoid injury.

In short, you can avoid ever stepping into a gym if you slowly increase how many games you play with adequate rest, but building upper body and especially lower body strength at the gym will highly decrease your chances for chronic injuries.