Yes, I agree there are other factors at play e.g. income inequality, education etc.
Also, how often the death penalty is actually used - not often in the US, more often in Singapore.
As I've said before, I'm not in favour of the death penalty due to the potential of innocent lives being lost, but I do think it can reduce crime.
The recidivism (reoffending) rate after 2 years in the UK is around 60%. In Singapore it's around 30%.
It makes sense to believe that the death penalty plays a role here. If a murderer is released after 10 years (many are released much earlier!) there's a much higher chance of them killing again than if that same murderer was executed after the first crime - making their reoffending rate 0.