The scientific definition of stress is any activity above basal metabolic rate, ie if the heart rate increases. So yes, the deer are stressed. The non-scientific have hijacked the term and equate it to distress.
The deer have to be culled. The only two practical methods are hunting and stalking. Hunting is a selective cull which is an organised, monitored and disciplined procedure carried out by hunt staff. The followers are the supporters who finance the operation. They go to watch the battle of wits between hounds and deer. The selected deer are the old, infirm, injured and slow, resulting in a healthier breeding herd.
Stalking is only selective in that the deer is in the wrong place at the wrong time. The stalker has the deer in his sights and fires.
It is not recommended that the stalker aims at the head - where death is instant - because the target area is too small and there is a possibility of smashing the jaw, the deer dying of starvation days later. The recommended area is the thorax or abdomen. The deer may or may not fall immediately. The advice is that it should be left undisturbed for 10 minutes plus (in case it is stimulated and runs off).
The bullet causes massive damage to the internal organs. The deer haemorrhages; when insufficient blood reaches the brain it becomes unconscious. This can take up to 10 minutes.
The above is the ideal scenario - an expert, well-practiced marksman under good conditions. With less-skilled marksmen or indifferent conditions the deer are more likely to be injured and escape, to suffer possibly for life. The hunt is frequently called in to find such deer and dispatch them. A conservative estimate of 5 per cent of the deer are injured by stalking.
With hunting there is an end point to the suffering, the deer being shot at the end of a hunt. Those that escape are tired but unharmed.
There are two sides to every story.
D J B DENNY BVet Med MRC.S