When the death penalty was abolished, women finally had the vote, slavery was abolished and so on this change was brought about to a greater or lesser degree by a relatively few people who for one reason or another believed enough in these issues to fight for them .
Some of them were then accepted with a resigned shrug by the general population, but certainly not necessarily by majority consensus.
No majority consensus for women to have the vote? I suggest almost 50% of the population were probably in favour ...
Half a million women and men demonstrated, at a single event, in favour of it in 1908. Quite a good turn out for something without public consensus.
Slavery? Well there were individuals like Wilberforce who campaigned vociferously, but I'm not sure the general UK population were either against his efforts or simply resigned to enduring the ending of it. In fact I'd say that because the vast majority of British owned slaves were in the West Indies the ordinary UK population probably had little idea of the atrocities being committed, or its scale. The more people knew, the more the consensus grew to abolish it.
Similarly, the death penalty. Public opinion on the issue moved because of a general growing uneasiness with capital punishment and a few specific cases, such as Derek Bentley and Ruth Ellis. There were no mass demonstrations to keep hanging. For all people say they favour capital punishment, it's often just frustration with what they perceive as lenient sentencing. Most of them would pick life without parole for murderers over execution in a straight choice.
If a government tries to impose a law without public consensus because it imagines it is taking a moral lead, there needs to be very careful analysis to make sure they are not simply listening to a vociferous minority.