This would almost certainly lead to a bunch of users making accounts on other sites for no other reason than to have an account and try to farm the badge.
There is a wide range of topics covered by Stack Exchange sites, and that range grows all the time. The chances of having users who truly cross that entire domain are very, very small. Understanding that 90% of all statistics are made up on the spot, I'd argue that well over 99% of the people with this badge won't actually use all of the Stack Exchange sites and, instead, just pollute those sites with an unnecessary user account just to get the badge.
One might take it a step further and say, "Fine, then you need to at least participate in the other sites in order to qualify for the badge." Except that this won't change the demographic of people who try to acquire the badge. It will only persuade them to post low quality content on sites in which they aren't actually interested.
In short, I just don't see this as encouraging or rewarding the kind of behavior Stack Exchange wants.