Hunters, in my opinion, are more active, better organized as a lobby, and also perceived publicly as more important lobby. Birders still are perceived, and see themselves, as a narrow niche hobby.
Fox hunting was enjoyed by the very elite of British society, Royalty, the titled, judges, landowners, MPs. If ever there was an organised lobby, they were it. Yet it got banned, not through nature lovers / mammal watchers winning the battle over the hunters, but through that innocent glass screen we keep in our houses.
There had been a growing view that fox hunting had become something medieval, like hawking, or bear baiting, yet the masses were willing to look the other way because there was also a belief that only old or diseased foxes got caught, and that the fit ones got away (a sort of unnatural selection). Then came a news feature, broadcast nationally, and said masses found themselves witnessing the filming of a hunt that had chased a fox to ground, before sending in a terrier to drag it out. Letters of complaint were written, a lot of letters, and the ball started rolling (incidentally, the bulletin hadn't been shown to highlight the cruelty, but because Prince Charles was a member of the hunt involved, and in the UK, Royalty + Controversy = instantaneous blanket headlines).
Of course, hunting with guns isn't hunting with dogs, and here the masses DO look the other way (prior to tipping off the British press, maybe someone ought to persuade Prince Charles to nip up to a Cumbrian Grouse moor and take out a Hen Harrier with a twelve bore). Dark humour aside, in these wildlife aware times, maybe hunting will dwindle away, maybe it wont. In the absence of some serious e-mailing, I suspect we'll hear the distant thump of gunshot for sometime yet.
The Hunting with Dogs Act, President Trump, Brexit -never underestimate the power of the many.