Some interesting explanations going on.

Like many things the reason why an item started are not always known and yes the tonsure could well have led to cold heads - maybe a good argument for the more folliclly challenged among us to wear hats!

I must confess that I have never myself seen a verger with headwear but then again many churches do not have vergers, at least ones with ceremonial or processional functions.

And yes there is a conundrum for pipers. It isn't easy to pipe and remove headgear at the same time and where could you carry the removed headgear anyway? It would be almost as difficult with a Glengarry as a feather bonnet so discussion and agreement beforehand with the clergy would be a good idea as Daz says.