I have the traditional canes,thought I would try this as well. :)
That will work, I do similar things with my peas. Just make sure its solid.
Runner Beans climb in a spiral fashion and need canes to twist round. The frame you have is not ideal for them.
The frame you have will be fine for peas especially sweet peas. They cling on differently and will attach to netting or the wire frame you have, beautifully.
Some varieties of courgettes, gourds and cucumbers like to climb and the frame will be right for them too.
To explain the 'not ideal' comment further, as you know with the 'traditional cane' method the beans spiral up the cane unhindered.
With your netting when they reach the horizontal wires they may want to spiral along it so you will have to be vigilant and tie them in to the verticals again.
As has been mentioned previously you will have to give them strong support. When covered in beans it will turn the structure into what is effectively a 'big sail' so it might want some guy ropes or similar to prevent it being blown over.
Other than that I see no problem!