If you don't cull then the older stags essentially die of starvation. Which is all very natural, I suppose, but probably not very nice for the stags.
As Emma says, there are no natural predators for deer in the UK, yet, land boundaries are finite. In the wild deer will migrate long distances to find pasture when the herd size increases, but it's not so easy in the UK (that, by the way, is why you see all those deer signs by motorways) so the only thing that regulates herd size is culling.
Most culling is carried out by professional stalkers, but 'trophy' beasts - those with reasonable antlers - attract a premium and the rights to shoot them are sold. The money that someone will pay to shoot a trophy stag will pay for a lot of husbandry and conservation on the estate. You can argue until the cows come home about the mentality of someone who wants to shoot a trophy stag, but it seems daft not to capitalise on the value of the sport for the betterment of the estate.
As for shooting during the rut; that's not usually done, and I wouldn't be in favour of it, but we don't know the full circumstances.