In 'Harrison Bergeron" society is equal, though not in a metaphorical way, but in a rather creepy way. The only thing that is the equality target are not rights, but intelligence; every single person is intelligent in the same way, or maybe, ignorant in the same way. In order to achieve this, people who are gifted - compared to others - get handicapped, so everyone is mentally the same.
It is pretty clear that a society like this can not work, because ignorance portrayed in a massive society-like scale, would simply damage the system.
So, would a society of not gifted people getting "lifted up" better? Well yes, but actually no.
In fact, compared to the distopian society of "Harrison Bergeron" it is indeed better; however, a society without diversity can not be called society, but it can be defined better as a "gigantic group of Doppelgangers"   E & V