I'm afraid I don't know much about it, I was just searching to see if my mum's Hoeysuckle was evergreen or not and came across the one in the link, so thought it might fit the bill.
White ( summer) Jasmine can go rampant. Very sweet-smelling flowers, too. But not evergreen.
But the blue-flowered ceanothus grows extremely fast - mine grew from seed to a 10ft diameter ball in about 10 years. This should be evergreen.
Unfortunately any quick growing climber is likely to become a problem eventually. It doesn't know when to stop being a quick growing climber you see! But clematis is so pretty its worth it and fairly easy to trim back when you need to.