With clipped wings they might be ok, but my Blue Rooster is a Blue Andulusion and he can hop up on the 6ft chain link with no problem. Nugget has so sit on the 4ft fence, he's too heavy to make the 6 ft. All my Bantams can make the 6 ft though.
I have a speckled sussex cockerel who periodically has to be chased down off the 6' high chainlink of their temporary run -- and it is only 4x10 so it isn't like he has lots and lots of room to fly up there either.
So it just comes down to what whims/urges/fears they happen to have at any given moment.
My 20 week old Silver Laced Wyandotte pullets regularly sail over the 6 ft plus fence around their pen. Even had a Barred Rock fly over 6 ft up onto the coop roof. Normally, they dont do that, but sometimes, they're just highly motivated.
I had two EE roos that always flew over the run fence as soon as I let them out of the coop in the morning. they stayed close to the run where the rest of the chickens were, so it wasn't a problem, but until I netted the top, I couldn't make them stay put. I also found a BR cockerel up on the coop (over 6 ft) one morning when the coop door blew closed before they went in for the night. you might be OK if they have plenty of running room, mine never fly over the 4' fence that separates the pasture from the back yard, but there's really nothing but dead weeds in the back yard, and they have nearly half an acre of run.
The bantams and the light weight standards can go over a 6 foot fence no problem.
You mention you are having a privacy fenced installed, I am assuming around your yard.
I can tell you from experience that I have a four foot chain link fence around a portion my yard which includes the chicken coop and my chickens a mixture of bantams, light and heavy weight breeds and they never go over this fence. The area is about an 1/2 acre. I am assuming they have plenty of room to range and have no desire to go over it. I have never clipped a wing.
I would see how they do, they maybe perfectly happy in yard.