Okay, let's look at this class with an eye for improving it and making make it tighter.
Aerial Artistry
Starting at 3rd level, you gain the ability to move with incredible speed, precision, and power. Few obstacles can prevent you from reaching your destination.
When you move, you can instead take two short movements by flying. Each movement is at half your speed, and you must end each one on a creature, solid object, or ground. If you do not, you fall and your movement ends.
The idea of this power is you can fly short distances. Mechanically, you fly but narratively you parkour up an otherwise unclimbable surface.
But, as noted, this doesn't work well when you CAN climb. While you can "fly" fifteen feet (half your movement), this also means you can't really jump farther than a Thief when you gain this subclass. And the Thief might actually be able to jump *farther* at higher levels. With just 12 Strength and a 20 Dex, the Thief can jump farther than you can fly; although the Acrobat has the advantage of not needing a running start...
The fix is easy: just give the Acrobat the climbing part of Second-Story Work as well. Aerial Artistry is neat by not game breaking, so you can pair the two easily enough.
Fearless Aerialist
Starting at 9th level, you no longer take damage from falling.
This is fine.
Unchained Agility
At 13th level, your agility, speed, and flexibility allow you to escape from almost any danger. You gain the benefits of freedom of movement.
This should probably just include the text of the abilities rather than giving the class a spell. But this was almost certainly a shorthand.
Aerial Mastery
At 17th level, your Aerial Artistry feature improves. When you move, you can instead take 4 short movements by flying.
Again, this is fine but kludge. It needs to be rewritten and rephrased.
If working on this for my own homebrew document or a player wanting this in my game I'd do this:
Aerial Artistry
When you adopt this archetype at 3rd level you gain the ability to move with incredible speed, precision, and power, letting few obstacles can prevent you from reaching your destination. Climbing no longer costs you extra Movement. Additionally, during your movement you can fly up to half your speed. You must end this movement on solid surface, such as a creature or the ground, otherwise your movement ends and you fall.
You can only use this feature twice per turn.
Alternatively, I'd phrase it as:
Aerial Artistry
When you adopt this archetype at 3rd level you gain the ability to move with incredible speed, precision, and power, letting few obstacles can prevent you from reaching your destination. Climbing no longer costs you extra Movement. In addition, twice per turn you can gain a fly speed up to half your speed, but must end this movement on solid surface, such as a creature or the ground, otherwise your movement ends and you fall.
Fearless Aerialist
Starting at 9th level, you no longer take damage from falling.
Unchained Agility
At 13th level, your agility, speed, and flexibility allow you to escape from almost any danger. Your movement is unaffected by nonmagical difficult terrain, and spells and other magical effects can't reduce your speed or cause you to be paralyzed or restrained. You can also spend 5 feet of movement to automatically escape from nonmagical restraints, such as manacles or a creature that has you grappled.
Aerial Mastery
At 17th level, your Aerial Artistry feature improves. You can use your Aerial Artistry up to four times each turn.