As you mentioned, yes, only Charizard, Aerodactyl, and Dragonite can be ridden in midair.
If you have not beaten the E4, you can still ride on them, but only close to the ground, giving you no advantages other than increased speed. After the E4, you fly much higher above the ground, allowing you to fly over most obstacles such as as ledges, NPCs, and water. The only exceptions are Routes 16, 17, and 18 (the routes that formed Cycling Road in older games), where you can fly at the increased height even before beating the E4.
If you’re in the air after the E4, you also have a small chance of encountering wild Charizard or Dragonite on just about every route. This does not apply to the Cycling routes before beating the E4.
If you use Sky Dash (the technique that replaced Fly) while riding a flying Pokémon, the animation changes slightly to your Pokémon flying you offscreen instead of the balloon chair.