I think garlic mustard seeds must need light to germinate or something?
We cut back a bunch of thistles and honeysuckle at the edge of our woods, and the ground underneath the honeysuckle was completely barren - until the next spring when it became absolutely covered with garlic mustard.
So your creeping charlie is only a temporary solution
As someone who has lived in garlic mustard territory for decades, here are tips:
-The first year from seed it is usually a short rosette with no flowers. So you have an extra year to pull it before it spreads.
-It pulls easily, but the flowers will still ripen to viable seed on an uprooted plant, so don't leave them lying around.
-It is one of the first things to sprout in the spring, so you can hit it with a non-selective herbicide before your other plants have sprouted if that's part of your gardening strategy.