I agree with your vet that most people overfeed pellets thru nothing but inaccurate and outdated information. We feed a very limited number of pellets here....some of our rabbits get no pellets at all, some of our smaller buns get only a tablespoon or so.....even our two bonded Flemish Giants get only about 1/3 cup pellets per day to share....and all have access to high quality hay. But....and this is a big but......we still have these animals (naturally wild) in an artificially created environment feeding them virtually an artificial diet. And the mention of wild rabbits in this thread.....internally domestics and wild rabbits are the same....and our treatment protocols, for the most part, are identical. So why can cottontails live healthy without pellets while out domestics maybe can't?
So, having said that and hopefully everyone can understand my rambling.....the question comes down to if you can feed rabbits without pellets. Certainly....look at wild rabbits. They don't eat pellets....and they are never obese either....and they are very healthy. They eat basically scrub plants....leaves and grass, and in the winter, they eat bark. All of these are very low in proteins and nutrition. So how do they do it? First, like our domestics, they are hind gut fermenters. This type of digestion lends itself to being very efficient. The reingestion of undigested cecals extracts all the nutrition possible from the low quality foods that wild rabbits eat. But wild rabbits also get something that most of our domestics don't. Unfiltered sunlight. Sunlight is one of two ways that animalsget Vitamin D. Vitamin D is essential in the absorbing of calcium and other nutrients. Vitamin D deficiencies can be found in many indoor species....mammals (including most humans), reptiles and birds. In the abscense of unfiltered sunlight, the only way to get Vitamin D is by dietary supplements...and that is where high quality pellets come into play. Fortified pellets, in a limited amount, will give the necessary supplements for a healthy life. Indeed, our use of pellets for our rabbits is strictly to maintain weight and body conditioning. The rabbits we have on hay only get Vitamin D supplements.
Vitamin D causes calcium to be lost and results in a condition known as Metabolic Bone Disease. But don't let the name fool you as it is a system wide problem in animals afflicted. Not only does MBD result in skeletal and dental issues, it can cause organ failure, seizures and death among other things. People are so caught up in the old and incorrect idea that rabbits should avoid calcium. even vets have jumped on that wagon. Fact is, rabbits have bones and teeth and those require calcium to be healthy. Many skeletal and dental issues can be traced to MBD. Ask your vet about Vitamin D deficiencies, MBD and the systemic problems caused by MBD. For those of us that do advanced work in wildlife rehab, we are keenly aware of issues from MBD. We see it often in squirrels, opossums and birds caused by lack of sunlight and nutritional deficiencies.
Now, the absolute answer to your question is absolutely YES...you can feed an all natural diet without pellets...as long as you provide everything that a rabbit would find in nature. And that includes exposure to unfiltered sunlight, the equal spectrums of artificial sunlight or provide a proper supplementation of Vitamin D.
Randy