If I had to bet, the perpetrator(s) is shooting the animals from their property or close nearby. That would give them a shoot and cover position, never to be seen walking trails with a weapon. Some parts of the green belts are rugged and have no trails other than deer trails, pretty thick where I've seen. Other parts of the green belts that have actual hiking trails, are more populated and you'd never get away with this without being seen.
I was going to suggest the same thing. The perpetrator is likely shooting from their own property, or very close to their own property, so trail cams are unlikely to solve this one.
The area I grew up in had several green belts running all through it, and they were FANTASTIC places to hide away if you had nefarious intent (or just to explore if you were a nature-loving kid like me). Plus I was VERY good at hiding away in them. People would pass within feet of me often and never know I was there if I didn't want to be seen.
All that being said, criminals aren't known for being terribly smart, and they can get careless when they think they're "getting away with it easily". So I might put up trail cams anyway? I'd be concerned about the legalities of putting up surveillance devices in public-use areas though...
If it were me, I'd be combing social media looking for someone complaining about the deer eating their flowers and crapping in their yard, and maybe start there. Maybe cross-reference that with people posting about hunting in the past, or asking questions about hunting or getting into the sport.
Might not help if it's a psychopath future serial killer, but it'd be a place to start.
I agree with the others though; as an avid outdoorsman and hunter, with a healthy respect for nature, those images are difficult to see.