We only cut our leylandii once a year, early autumnm, no problem, when we did miss it was a problem the next year but took the hacking back quite well.
Otherwise just give it a hair cut, about 2ft from the ground, should make a great bonfire in November.
But for short term shelter or if you're prepared to be the boss I don't think you can beat it.
we also planted some laurel, I find it comes out sideways too much, it's recommended to use secateurs to trim so you don't get damaged leaves and possible die back, with the leylandii we just use either the shears or petrol hedge trimmer.
I also have to be absolutely sure the goats don't get anywhere near the laurel, they sometimes grab a mouthful Leylandii but a little won't harm them (the kids got loose and ate more than a mouthful but were OK).
Saw a video on Buttercups goat sanctuary and they actually FED Leylandii to the goats!
SF - why not take alternate plants out, improve the soil and start something else going, eventually replace it all? I believe they are shallow rooted so they shouldn't be too bad.
There is a tree nursery in East Lothian which supplies reasonably priced trees grown in rootrainers that get going easily. )eg hornbeam (in 15s) 70p ea, or of you get 105+ = 46p ea
Actually - hornbeam, that's another one to throw in the suggestions