I believe what was meant to be said is they're totally herbivores. Not omnivores. No meat for them! No bugs, and very little high protein with low digestability like soy. Even corn is bad.
I would never EVER put a rabbit free-ranging with chickens/turkeys. There's a lot of health issues that will transfer from birds to rabbits on the ground that are easy to treat in birds, not so in rabbits. I've heard horror stories of the animals trying to mate eachother and being injured. A doe will try to kill a bird near her nest and the birds may very well try to eat the babies, or when a bird gets big enough even a small adult. Also the rabbits sometimes snack on the too-high-protein turkey feed or the too high calcium chicken feed. This can cause some serious gut issues and bladder issues and could kill them. But if you're feeding FF most rabbits won't want it. Rabbits should be eating a low acid diet since high acidity in their gut also causes problems so no fermented feed for them (although some ACV in the water is good, just keep it low quantity). The digestive tract of a rabbit is so violently different from a chickens. Compared to a chicken their diet seems super sensitive... But the truth is they're just good at eating certain things and nothing else. In this case, grass. Chickens eat grass too but it only makes up a small amount of their diet compared to all the bugs and grains and weeds etc. For rabbits it's the other way around. Mixing 5-6 different types of grass is an adequate diet for a rabbit, even one having a couple of litters each year. For high production (4-10 litters a year) they need more than that. So grains and "vegetables" like oats, dandelions, barley, plantago, wheat, alfalfa, clover, carrots, beet tops, turnip greens, chard, spinach, arugula, romaine, sow thistle, peppers, even sometimes corn etc. They make a big difference. Otherwise you wear out your rabbit, they may not produce enough milk for their kits, kits may die out, does will loose condition, etc.
I'd just like to point out that wild rabbits work with wild food the same way wild chickens do. In general a rabbit in the wild will raise 2-5 kits a YEAR. A forage diet will NOT cover our high production (up to 12 kits a litter 4+ times a year, so up to 20X's the number of kits) rabbit just like you can't get away with feeding free range laying hens without added calcium or high calcium soil/feed sources. You may get two decent litters out of a rabbit on forage while maintaining good condition, but not the numbers one normally hopes for. You'd have to plant a very specialized pen of forage to get by with free-ranging and high production standards.
Also, fencing does not always work for rabbits, regardless of types. They dig. And dig and dig. They're burrowing animals. They will tunnel 20 feet to get out of a pen and you will never know it if they have to. Some rabbits are content to sit in a pen. Some of mine are happy in a pen. Some dig a foot deep hole if I leave them out for more than six hours on the ground. And rabbits raised on the ground often have serious parasite issues. They're just more sensitive than chickens in every way. You can't even use many antibiotics on them or it could kill them, and if you don't work extra hard to maintain gut flora any antibiotic could kill them. But good management seems to prevent most of those problems and you can have happy, healthy, natural rabbits even in cages.
I've heard free-ranging rabbits works best in very dry environments where ground based parasites are lower in number. It works best in a huge pen with a colony setting where you care less about numbers and sires and keeping them from fighting, and more about getting to eat rabbit sometimes and having fluffy bunnies.
Man I sure am learning some stuff here lately and eating humble pie. 
