Sure, innate magic seems pretty similar. But, like you pointed out and like was pointed out by the designers, 5e doesn't care about that. You cannot affect Dragon Flight or Roc Flight in the game, whether it is mundane, supernatural or innate magic doesn't matter. You can cast a spell to ground people, but that would apply regardless of the source of the flight.
Same with Paladin Auras, Monk Ki, Barbarian Rage.
And that is 100% of the problem with Anti-Magic. What is a magical effect?
A Druid's Wildshape says that they take the form of an animal "magically" so it could be argued that Anti-magic suppresses it. A Monk's fists "count as magical" for overcoming resistance, but by that phrasing, they are not magical themselves. They just count as magical. A Paladin's Lay on Hands is because they have a "blessed touch" and it never says the word "magical" once in the entire description of the ability, same with Aura of Protection. Meanwhile, the Immunity to disease is because they have divine magic flowing through their body.
So... could a paladin catch a disease while in an Anti-magic sphere, but they could lay on hands to remove it? Because one says magic and the other doesn't. What about the aura of Sunlight the Devotion Paladin can emit? It doesn't say it is magical in any way.
But, it can't just say "spells" because magic items don't always create spells, and that is the trick. Anti-magic fields were designed to care about spells and magic items... including Adamantium Armor which isn't enchanted, just made out of adamantium.
So, you have a problem. What is "magical"? Is the ability for a dragon to fly and shoot acid magical? What about a Paladin's aura and ability to heal? What about the protective benefits of Adamantium? The Shadow Monk's teleport? We absolutely want to make sure Dragons can still fly and live in an anti-magic field, because anti-magic isn't meant to be a death sentence against dragons, undead and giants. But if a Lich can still exist in anti-magic, why can't my paladin's aura? And if the paladin aura can because it doesn't say it is magical, why can't my Shadow Monk still use CLoak of Shadows to become invisible.
Magic ie Casting spells and spell like effects is the easy part of all of this. The rest of it... 5e didn't care about, because it wasn't something meant to be cared about.