As a sidesaddle rider, I'm not sure you can include that in this hypothetical debate - as the cane is a very different tool from a "whip" - and used in a very different way.
Sorry but the BD rule book clearly states that whips are allowed at ALL levels, excepting Championships when they are not allowed at any level. You must be referring to Pony Club and Eventing rules.
I've been brought up on the fact that one tap of the whip is better than 20 successive digs in the ribs with your heels. Further to that, the whip is often used to teach a horse to move off the leg. So, add on that it's s safety measure in e.g. XC, I really don't think it can be banned from racing, XC or SJ.
A whip can however be misused: this is precisely what the regulations are to protect against, helping competitors to find the balance between using the whip correctly for genuine support, and battering a horse on the assumption you need to in order to get the best out of it.