A lot of people want the ability to go "invisible" with the Battletag system. That is, to appear offline to everyone else, but be able to see who's online.
I sympathize with this desire. Sometimes you just want to play by yourself in peace. But it feels socially awkward to request that solitude when other people can see you online. As well, a lot of people don't respect the Busy status. They figure that means you are Busy to other people, and that you'd be fine if they contacted you.
However, does no one remember ICQ and the late 1990s? The arms race that was "I am invisible to most people, but these specific people can see me if I'm invisible, unless I'm really invisible." Honestly, the entire concept of invisibility just made things overly complex.
The thing is that in an ideal world we want things to be asymmetric. We want perfect opaqueness for ourselves, and perfect transparency for others. But this cannot work, because applies to everyone. There's no point to a friends list where everyone is invisible.
I think that a better option, instead of invisibility, is to allow the option to "go dark" as a status. No one can see that you are online, but you can't tell if any of your friends are online either. To others you would appear offline, and everyone in your list appears offline. This would preserve the symmetry of the Battletag relationship. It gives you incentive to appear online as the default so you can see if your friends are online. But it would also allow you to play privately by yourself in peace.