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As you mention, it's about rhythm and timing with your stride to maximize comfort and minimize effort so that running is challenging but not unnecessarily strenuous.

Being conscious of your breath forces you to manage pace. If you're trying to maintain an aerobic pace, noticing when you start breathing hard is a signal to dial back.

Additionally, particularly in colder climates, it's also helpful to breathe in through the nose and out through the mouth, in order to warm and moisten the air before it hits your lungs, thus increasing comfort.

Sounds like you've probably already landed on an effect breathing technique for you. But just because you figured that out on your own, don't assume everyone else naturally will. People unfamiliar with aerobic exercise may not realize it's abnormal to be gasping and out of breath while running.