Great idea!
In the beginning I thought chicken wire isn't suitable as it's not strong enough, but as you said, with some additional vertical support and maybe overlap several layers, also make the tower a bit lower, chicken wire should work.
I kept thinking how to sow on such tower next growing season, as the planting areas are vertical and small (no matter the chicken wire or hardware cloth), sowing tiny seeds will be comparatively difficult.
Seed bomb is one solution, another idea I have now is using the pastry bag, just mixed the soil, water and seeds, and we can somehow "inject the liquid seed serum" (OMG, it sounds so funny) to regenerate a green food tower.