Buy dragon ledges. They mount to your cage then all of your branches have a place to attach to.
https://dragonstrand.com/dragon-ledges/
There is nothing like what you are talking about. I would advise against using push pins as well unless you can do two into the branch end so they do not roll. Additionally you can use plastic lattice inside your cage. Mount branches and vines to that. you would zip tie the corners of the lattice so it slides into the cage against the back and two side panels. Then your mounting the branches and vines on that with zip ties. Clean cut your zip ties so there are no sharp pieces poking out.