Firstly, you can't remove an authenticator via phone support. They just WON'T do it. I think my friend had success with trying it that way ONCE out of 52 attempts. So not to denote your creative thinking. But it does take quite a bit less info then you might thing to achieve this. No phone call needed and if you want a really easy way to remove it yourself... It just takes some really creative social skills.
I will agree with you that an authenticator is an extra "layer" of security but it won't detour those who know how to get around it with little effort