You are right that nothing proves that droping penal death reduces the crimes. There could be no link between both, it could be a pure coincidence happening in the hundred countries droping it.

Also, and despite a constant populist pressure in each of this countries, very few have reintroduced it (and most of them fastly droped it back again).

Of course, some may have droped it out of diplomatic pressure more than for their population… but still, it brings benefices. Sorry, I mean the coincidental effect on crimes is observed ; and so at the very least droping penal death does not rise the numbers nor severity of crimes.