Pole beans will not climb horizontal wire. Peas will, because they have tendrils that grab almost anything while he vine grows straight. Pole beans wrap the vine around what they are climbing, and won't grow horizontally. You need vertical poles, string, or wire for pole beans.
Well for two years I used 2X4 fencing for the peas and pole beans and it worked well except for cleaning at the end of the season.
I have gone to livestock panels this year. heavier and easier to clean. See my Arch Of Beans thread.
It takes a little training to get them started up the fence then they climb without a problem.
As you say they don't follow the horizontal wire but climb the verticals.
Sure is a lot nicer than tearing up the back stooping to pick beans!! Seems as I get older I find it more difficult to get back Up.
What's up with that??