But the government doesn't have to pick a side. Or if it does, there aren't necessarily only two sides to pick from, at least one other available side is the side of the freedom of individuals to choose.
For example - to take something relatively innocuous and uncontroversial, at least in our country, at least in our time - the government does not have to be either for or against Catholicism. This wasn't always the case, but we are way better off now that the government no longer dictates one's religion. Similarly we are way better off that the government no longer dictates who we may or may not marry (or not much anyway).
(Death penalty is a different case, because it is explicitly a government power to be either exercised or not exercised, so it has to have a position on that.)
Obviously there are all sorts of exceptions and nuances and special cases, and obviously there's a sort of Overton window where things change over time. But I think it is stretching a point to say, for example, that foxhunters are imposing their morality on others if foxhunting is permitted - so long as it isn't actually compulsory.
In general I come down in favour of letting people do what they want unless there is a compelling reason otherwise.
While what you're describing sounds like some form of American Classical Liberalism (let people do whatever the hell they want, just about, and keep the government far away), sometimes Governments do make judgement calls on what posters in here are referring to as "optics". How it looks on the country, what message it sends, what it does to masses of the general public and then tie that up with actions. What are the benefits? What is the process? How is it regulated? Who suffers if anyone?
Considering this is revolving around death, because, yes, it is the killing of something for sport, it is going to be a contentious debate around both optics and actions. For the most part the sports our country likes to take part in, while competitive, do not function around the goal of killing. Horse riding which uses animals gets some flak, but that is more about the treatment of the horses, notably at the grand national when there is definitely overcrowding which is liable to cause accidents. The balance of human behaviour around wildlife is always up for debating. We do enough as it is to take from the planet, and everyone's friend David Attenborough travels around the planet explaining fairly well why at times we just need to leave wildlife the fuck alone if we can.
Fox-hunting just doesn't have a "friendly" outcome. The only outcome is death and suffering. It's not even humane killing. The death itself not falling into the categories I outlined earlier in the topic around man having to accept the protection of livestock/crops may result in self-defence. When you start going out of your way, into wildlife's homes and killing them for sport, there is going to be "compelling reasons" for it to at least be debated. The foxes aren't even used as a source of food. It's a waste of life for... people to laugh/cheer?
Then the foxes aside, we also need to consider the well-being of the dogs which are pretty much weaponized. Often in inhumane ways which attempt to make them overly vicious and bloodlust. Starvation/beating and inciting them to show aggression/aggressive traits on demand.
There's a lot to unwrap and thankfully most of the country, MPs included, understand there is a fuck load more the UK can be proud of or put forward as the sports/interests of our nation without scrambling to idolise a blood sport. Both with optics, and thinking about the actions of the people wanting to do it and the potential harm caused. Some things are just better off left in the past.