RAW : You do not break the grapple
Let's take a closer look at the text you've mentioned for grapple rules (emphasis mine) :
The condition also ends if an Effect removes the grappled creature from the reach of the Grappler or Grappling Effect, such as when a creature is hurled away by the Thunderwave spell.
So if an effect moves the grappled creature, they will break out of the grapple. So, is the effect from Tempestuous Magic moving the character away from the grappler?
Here's the rule text you cited (emphasis mine once again) :
Starting at 1st level, you can use a bonus action on your turn to cause whirling gusts of elemental air to briefly surround you, immediately before or after you cast a spell of 1st level or higher. Doing so allows you to fly up to 10 feet without provoking opportunity attacks.
There are two things to note here. The first is that the effect does not move you, but rather gives you the ability to fly for a certain distance. While it is worth pointing out, this first argument does not weight very heavily in the balance, especially since usually bonuses to movement speed are explicitely indicated as "speed".
So, at this point this "flying" could be just about anything. However, the second point weights a little more in the balance : you do not trigger opportunity attacks with this movement.
The rule text on opportunity attacks states the following (you guessed it, emphasis mine) :
A creature doesn’t provoke an opportunity attack if it is moved without the use of its movement, its action, or its reaction. For example, the effect of the antipathy/sympathy spell requires the target to use its movement, meaning that it would provoke opportunity attacks when it does so.
Similarly, dissonant whispers requires the target to move using its reaction (if available), so that activity also provokes opportunity attacks. In contrast, a creature that’s pushed by a gust of wind spell does not provoke opportunity attacks.
If the fact this does not trigger an opportunity attack had to be explicitely stated, then this kind of flying movement must fall into one of those categories.
- It is not an action. The closest you could say is that the effect is triggered by the use of a bonus action, but even then that's not an action.
- It is not a reaction.
This only leave movement. Following this reasoning, it seems logical to conclude that the effect of Tempestuous Magic does not move you itself, but instead temporarily grants you a tiny bit of flying. Or it would, if the rules for the grappled condition didn't prevent it from happening :
A grappled creature's speed becomes 0, and it can't benefit from any bonus to its speed.
In conclusion, looking at the rules alone, you cannot benefit from the flying effect of Tempestuous Magic while grappled. Of course, a DM may rule otherwise (after all, why would wind magic strong enough to make you fly not free you from a creature grappling you?).