Every time this comes up, the authenticator issues comes up as well.

It's really this simple:

If you are willing to lose EVERYTHING on ALL of your accounts and have to jump through hoops calling blizzard / possibly proving your identity to them (faxing them your driver's license or state / gov't ID) and waiting for GMs to restore all your stuff and HOPE to whatever sky fairy you pray to that they get it right the first time and you don't have to repeat the process because they forgot to give you something back that was rather crucial...

Have fun saving your $1 (mobile phone authenticator cost) or $6 (Keyfob cost). I hope your time you spend (should you ever have to) calling blizzard and getting all this sorted isn't worth more than $1 or $6. Personally, my time is MUCH more valuable.

Go cancel your homeowner's or renter's insurance as well. Why bother, right?

An extreme example, but it's the same point.