Why does the Catcher's Balk rule exist?

Not strictly true. An illegal pitch is a balk with runners on, a ball with no runners on. Not all balks are illegal pitches.

Example of a balk that’s not an illegal pitch: (1) failure to come to a stop. (2) Not stepping towards the base when throwing over. (3) Catcher’s balk.

Example of an illegal pitch (balk with runners on, ball without): (1) pitching without the foot on the rubber. (2) A quick pitch.

To be fair, McCarver was catching 70 years ago.