For example in the UK 58% of people believe that the death penalty should be allowed for some crimes (e.g. terrorist attacks). Only 32% oppose it (presumably with 10% undecided):

https://yougov.co.uk/topics/legal/articles-reports/2019/10/0...

So far from being the majority view, often anti-death penalty stance is the minority view but the political elite suppress it.

Let that really sink in, most of the comments here are very wrong in thinking the debate is over, with twice as many of the public still supporting it in a country where it's been abolished for over 50 years. Always remember to fight against capital punishment, the deal is not done.

I believe they do this as they understand the nuance better and realize that overall it causes more problems than it solves, so don't want to open that can of worms once it's shut. Looking back in history there's also significant political fallout every time someone is found innocent after their execution. Some hard-right politicians will band it around for easy points with their base, plus obviously the wider public too for more extreme crimes.