Another question had raised this issue when fairies were part of Unearthed Arcana and indicated that they couldn't because the flight was magical.
However, I noticed that on DND beyond that the flight is because of the fairy's wings, a non-magical source, and now I don't know what to believe.
The description of my flight from DND beyond says:
Because of your wings, you have a flying speed equal to your walking speed. You can't use this flying speed if you're wearing medium or heavy armor.
The phrasing, 'Because of your wings,' makes me think the flight is now non-magical.