TF2 hats used to bring an advantage until about a decade ago.

That is simply not true. Hats have always been cosmetics only.

Some unusual hats even give you a disadvantage as they broadcast your position through sounds.

This video describes how, in the past, certain item sets, which included hats, gave stats boosts when worn together:

https://www.youtube.com/watch?v=DZDIqnS0FcI

The author is completely ignoring that you didn't have to BUY these hats. For example, you could (and still can) craft the Milkman hat with 1 refined metal + 1 special delivery weapon.

All items had stats / skill changes, no?

Oh - were they already tradeable ?