I'd like to throw in a few facts at random:
Traditionally, the reason the majority of hunt staff "dressed up" in red coats was so that a farmer, seeing a group of riders on his land, could see at a distance and at a quick glance that they were the hunt and so riding there with permission, rather than any random group of pleasure riders who might not know and follow the rules of riding across farmland, crops etc. Many hunts have moved away from wearing red in recent years.
A trained marksman with a rifle would be a more humane method of fox control. Unfortunately it isn't possible to guarantee a clean kill shot 100% of the time - particularly not with a shotgun, which is the more common type of weapon used - so a percentage of all foxes shot will not die a quick death but a lingering one of starvation or infection. One advantage of fox hunting in that respect is that the hunted fox either gets away unhurt or is killed outright. There is no half measure, unlike every other method of control to a greater or lesser extent.
There is a misconception that foxes are running flat out, terrified, for hours. This is almost never the case. The fox will usually be far enough ahead of the hunt to move at a purposeful jog, quite calmly, pausing to sniff the air, assess the pursuit and pick its escape route. They are not chased for hours in a blind panic or anything near it. Only during the final short phase of the hunt, when the hounds are within sight of the fox, will it break into a run. Even then, running from a predator is 'natural' IYSWIM. It is far more stressful for a fox (or any wild animal) to be restrained than it is to be chased. That's not to say the fox won't feel a degree of fear during the end-stage of a hunt but only to a level consistent with its natural lifestyle. There is a tendency to anthropomorphise the thought of being chased as we humans would find it terrifying as it's such an alien concept to us, but it is much less alien or terrifying to a wild animal who lives on its instinct.
Far from being bloodthirsty slavering loonies hell-bent on torture, hunt staff in particular and many followers have respect for the fox. It is an unsentimental pragmatic 'old-skool' countryman's view but they neither hate foxes nor take perverse glee in watching it be 'ripped apart'. They see themselves, in the main, as having a job to do - a job which happens to give members of the community the opportunity to watch hounds at work, to ride an unknown line over open countryside and to strengthen social ties. The vast majority of people who followed hounds either did so because of enjoyment of the skill of houndwork - and there is skill involved; you have to be able to judge scenting conditions, ground conditions, predict the behaviour of the individual hounds which make up the pack, read subtle signs regarding trails, spoors, dens - or because it was the only opportunity to ride 'into the unknown'. Every other equestrian pursuit involves having the course or event entirely mapped out for you. Drag-hunting mimics hunting freedoms to some extent but even then, a person has previously chosen the trail based on a relatively restricted amount of land and there isn't the same unpredictability of route, speed, number of stops and starts, distance of runs etc that fox hunting offered.
A large number of people who hunt will have never seen the actual kill. That is not their primary motivation. It is the thrill of riding at speed over land that they would not otherwise have access to, because access was only granted on the basis the hunt was providing the farmer with a service. It may not have been the most efficient service, but it was a symbiotic relationship.
The ban was never about animal welfare. The independent enquiry set up to report on hunting during the parliamentary debates leading to the ban could not definitively recommend a ban on the grounds of cruelty as there was insufficient evidence to do so. The Labour Government had to invoke the Parliament Acts of 1911 and 1949 to pass the Hunting Act into law; only the seventh time it had been invoked since its inception.