The terms "attenuated" and "cold adapted" really need to appear somewhere in this discussion.
The strains that are shed by a person who has recieved the live vaccine are the same ones that were in the vax. They have been tweaked to make them unable to replicate efficiently enough to produce serious infection (attenuated), and to make them poorly tolerant of the higher temperatures found in the lower respiratory tract (cold adapted). Reassortment between the strains in the vaccine would not solve these problems for the virus. A mild form of the illness is not a side-effect of the vaccine, it is precisely the desired result. The idea is to give the immune system an opportunity to develop antibodies against viral antigens without the risk of serious complications from full-blown infection.
The concept shouldn't be that much of a stretch for anyone willing to consider deliberately exposing their child to, say, chicken pox -- the main difference being that in the latter case, nothing insures that a given child will not be among the small number who will develop serious complications (such as pneumonia, or encephalitis, or eye lesions leading to permanent blindness).