Hanging, drawing and quartering was the absolute worst execution method inflicted on criminals at the time and was only used when the crime was treason. It was only used on men; women accused of treason would be burnt at the stake, as would heretics. I think one of the reasons it was used only on men was because part of the quartering involved castration - no where near as much fun to watch if it's a woman being done!
There is some debate about the 'drawing'. It's position in the punishment title suggests it's after hanging and therefore means being 'drawn down' from the noose. But it might also refer to being 'drawn' to the place of execution - which was being pulled there on a trestle, usually upside down. Being hung was to suspend you mid air between heaven and earth, as you were unworthy of both. It was the quartering that killed you. But slowly. Hanging by the neck until dead was very much the kinder (and quicker!) option.
Crimes apart from treason or heresy would merit hanging or beheading. If you were very lucky and were convicted of either of those two crimes your sentence might be commuted to one of the easier deaths (Like Lady Jane Grey should have been burnt at the stake, for treason, but was beheaded). Only the upper upper crust of aristocracy would be executed in private - most executions were public, and a jolly day out. It wasn't until late Victorian times that executions were done inside prison walls - and the general public were not happy about the change in law!
If you think about circuses in Roman times all the way up to public hangings in Victorian times - most of human history has included public execution as fun, family viewing. If the government hadn't changed the law, we'd probably still be watching them now! I think this is very much a case of the law changing and people's sensibilities following on after, rather than people's views evolving and forcing a change in law.
(Case in point - we still happy to watch brutal and grisly executions for entertainment if we know they're only pretend! we still find the violence entertaining.)
But the past is a different country, they do things differently there - it certainly isn't 'this country' that was barbaric, more like the whole of our species.