No, you don't have to have a rooster. Hen-only flocks do just fine.
You may occasionally have a problem pullet/hen, but typically they're fine with enough space and resources.
You also just might not have a very good rooster.
My main rooster is (in my current opinion) great. A bit lazy when the hens are calling for him to check the nest box, but he's relatively gentle with them and always has been. None of the hens or pullets have lost feathers because of him.
Now, the cockerel.... bad bird. The gals started losing feathers when his hormones came in. The exact same birds that had never lost any feathers to the rooster I mentioned.
He got dispatched and nobody misses him, and feathers are already filling in, and they're still with their original rooster.