I think your question is based on an incorrect assumption that testosterone directly and solely causes aggression and other effects. I did a very cursory search for some research papers on the topic that immediately suggest things are nowhere near that simple.

Based on that, if you further tamped down any effects hormones have on human behavior and psychology, I think what would happen is that the many other environmental and genetic factors that cause anti social aggression, obsession, libido, etc., would continue to cause nearly as many problems as they already do.

If you want a better world then I personally think we need to work on eliminating childhood neglect and abuse and improving the emotional / mental health of parents (and, really, everyone).