Keep the plastic netting out of your soil. The least of your concerns is chickens having a hard time pulling root balls up. It will be keeping them from eating the plants from the top down. And speaking as a very serious gardener, (and I mean serious,) the last thing you want in your soil is a lot of plastic garbage that will cause you problems next season as you prepare your garden. The crap will be there for ever. ("Poop?" This website will not let me say "c r a p??" Who knew??)
Lots of people on lots of websites will tout the chicken as the ultimate insect scavenger for your garden. I suspect that they have read this somewhere and merely repeat it without benefit of actually having chickens ravage their garden and eat every green thing in sight.
Chickens love to scratch and any mulch on your raised beds will become mulch in the paths. Any cool, dry soil in the beds will become a recessed dust bathtub. Those sugar snap peas that you are letting dry on the vine for seed is the ultimate in chicken candy for these greedy beggers. Though you don't mind if they eat a few tomatoes, you will mind when they sample every one they can reach. Any seedling, if not eaten, will be scratched out, trampled by them or crushed by you as you scurry to chase the darn birds out of the garden. (Oh, I see it's "darn" birds as in "water over the darn." Too funny!)
So, after all these posts, it probably is pointless for me to say keep the chickens out of the garden until the crops are done. Been there, done that, got the bone-headed T shirt.
Wayne