It’s not always that simple. Sometimes metagaming can be subconscious, or just hard to avoid. It’s not always “to get an advantage” in such plain terms, and kicking players off the table for doing this on occasion… well, you do you, as always, but I won’t be surprised if you have difficulties keeping a table full when you DM.

Exactly.

The mage casts Chill Touch, just to spite you.

It’s probably because whenever spellcasters are being compared, a Wizard comes in and says something like “we’re the only ones who had to work for our magic”, as if finding a powerful entity and convincing it to make a not horribly skewed bargain, learning to commune with the land itself, or following the orders given by a deity isn’t “work”.

And don’t Bards also have to study their magic?

No, that’s different characters playing the same cast.