A .410 will kill a wood pigeon easily within its appropriate range.
It sounds to me as though your fellow allotment holder is a Walt - that "pathetic bird shot" is carrying about 600 ft lbs of energy at the muzzle - as opposed to the 12 ft lbs from an air rifle. Now the shot spreads, but even a quarter of that shot at thirty yards will be carrying far more enery than a pellet.
If its vermin control, you use the most efficient tool - Ed Cook demonstrated the efficiency of a .410 on an estate with a significant rabbit problem
As he puts it
Red
It's incredible the kill count some people can achieve, they must be seriously skilled in hunting vermin as a hunter myself I appreciate how difficult it is to stalk these little critters. And rik_uk3 I would imagine bunnies are more susceptible to birdshot than birds as they do not posses the thick coat of armour-like feathers to protect them. I swear even with my 11.6ft lb HW80 if you don't hit a pidgeon precisely in the neck or head, even at ranges as close as 8 meters he will survive and fly away as long as you havn't hit it's wing. Feathers are surprisingly tough! I believe 12ft lbs is close to 500ft/s of energy... so not massively different to a shotgun!
So whats the effective range of a 410? I recall my friend telling me he had to be almost on top of them to ensure a clean kill, pidgeon that is! I bet you could kill an bunny a lot further away than a pidegon.
As you can probably tell I'm not a fan of shotguns at all and would love to upgrade my HW80 to a silenced .22 LR however I could never justify owning one as I don't have land of my own and my little permission is just that... too little! It took me nearly two years to become familiar with all aspects of air rifle shooting/hunting, and about £400 to get the second hand HW80 shooting sweetly with a nice s/h Leupold sitting on top. With real firearms you just point and shoot... air gunning is easily the most difficult and skilled form of shooting there is, which I find nothing but annoying to be honest, however once you have a tuned and zeroed setup you are good to go.
p.s have you heard of the single shot 9mm parabellum rifles, innocuously referred to as "backyard guns" by many?