This is a great idea! The…

This is a great idea! The falconry community as a whole is extremely responsible in animal care and cares deeply about the wildlife they work closely with. This increase would not affect the the wild populations of birds. When a falconer traps a young bird and gives it experience hunting for a year or two or three, in a safe manner (the bird gets fed whether it catches the quarry or not, unlike in the wild) and then releases it they may actually help increase the population. It's like giving these young birds a learning opportunity with a safety net and vet care.
Many of the young birds trapped by falconers are covered in feather parasites like mites and have internal parasites like worms. Falconers get them to a vet to fix these problems and make them strong hunters prior to release. Statistically, 75% of red-tailed hawks die their first year - it's a steep learning curve in the wild. So trapping these young birds is more than likely giving a second chance to a bird who would not have made it without help.