Another interpretation of the data, also just a theory but no less valid, is that handguns rose in popularity as the weapon of choice within gangs, and the police responded by targeting gun crime. Again, I wouldn't want to state it as fact, but it would be a useful starting point for additional research.
The reason frankly why I would need so much convincing that banning hand guns had any effect gun crime rates in the UK is because I just don't think there were enough gun owners to be statistically significant in the first place. Handgun shooting was a niche sport - a very niche sport.
As for cherry picking: right back atcha. But even that doesn't work for the Ireland, Jamaica, DC, and Chicago graphs.
As for Ireland and Jamaica, the increase in homicides are very explainable given the Troubles and rise of Carribean drug trafficking in the 70s.
https://en.wikipedia.org/wiki/Simpson's_paradox
The chart is very clearly downward, with a couple of upward slopes in the middle. If you cherry pick the evidence like that crimeresearch.org link does, you'll get the upward slope. If you look at the data as a whole, you get the long term downward trend. This is pretty much the definition of Simpson's Paradox. The downward trend is the correct one.