The problem of "sports hunting" is that the hunters have interests that go against any sensible nature conservation - and they are pushing these interests hard in politics. So we have this absurd situation, where the hunter groups feed the deer in the winter, only then to cry how "overpopulated" they are and that they need to shoot them. And they even have the audacity to ask for subsidies, because the feed is so expensive! What is worse, they classify large predators as "nuisance animals" and shoot them. We now - just this week - have a motion to allow the shooting of wolves, because they do too much "damage" on the precious deer stocks (that, I remind you, have to be always culled due to terrible overpopulation).
What I find personally the most absurd is how the entire landscape is turned into a shooting range for a couple of months of the year and how the shooters act like they own the forests and that everyone else is just in their way - often quite aggressively. Note that in the Czech Republic, there is legal freedom to roam in any forest ... but they don't seem to care. I think that if this "hobby" did not exist and someone proposed to make it legal - such person would get laughed away very quickly. But because it is a "tradition", it's immovable. Well, because of that and also because about half of the MPs are active hunters ... That's really what you get for constantly voting in old men to govern you.
So yeah, this is not a "two sides with equally valid points" issue for me. One side cares only about their hobby, actively hurts conservation of many species, turns half the country into an overgrazed wasteland where people cannot walk without fearing to be shot ... and the other side just wants this nonsense to stop.