Would the Gust spell be able to move someone under the effect of the Levitate spell?

I am playing an air genasi storm sorcerer. Air genasi have the Mingle with the Wind racial trait, which lets them cast the levitate spell once per long rest (with no material components).

The description of the levitate spell says:

One creature or object of your choice that you can see within range rises vertically, up to 20 feet, and remains suspended there for the duration. The spell can levitate a target that weighs up to 500 pounds. An unwilling creature that succeeds on a Constitution saving throw is unaffected.

The target can move only by pushing or pulling against a fixed object or surface within reach (such as a wall or a ceiling), which allows it to move as if it were climbing. You can change the target’s altitude by up to 20 feet in either direction on your turn. If you are the target, you can move up or down as part of your move. Otherwise, you can use your action to move the target, which must remain within the spell’s range.

The description of gust states:

You seize the air and compel it to create one of the following effects at a point you can see within range:

  • One Medium or smaller creature that you choose must succeed on a Strength saving throw or be pushed up to 5 feet away from you.

  • You create a small blast of air capable of moving one object that is neither held nor carried and that weighs no more than 5 pounds. The object is pushed up to 10 feet away from you. It isn’t pushed with enough force to cause damage.

  • You create a harmless sensory affect using air, such as causing leaves to rustle, wind to slam shutters shut, or your clothing to ripple in a breeze.

I wanted to use the gust cantrip to move myself 5 feet in a direction, but my GM said it wouldn’t work, as levitate only allows the target to move by physical means as the spell states, and I couldn’t cast gust to target myself anyways.

I didn’t argue at the time, but having reread the spells, I’m not sure why it wouldn’t work.

Is it possible to cast levitate on yourself, then use gust to move yourself 5 feet?