I agree that introducing new foods slowly is a good plan. We did an experiment last year--we've always kept our rabbits indoors, sometimes in a colony, sometimes in pens in their own rabbit barn, but always in a building. Last year we tested several rabbits outdoors in the lawn (no, we don't spray any chemicals) in a rabbit tractor we moved daily. No hay. No pellets (they did get a little of an oat/sunflower seed/lentil/Calf Manna mix as a treat). I expected it to be a disaster (they were angora rabbits, and generally long hair and morning dew don't mix). Instead, they were happy and healthy, and grazed until December, despite a few snowstorms (we did give hay on the days the snow covered the grass). Their coats looked just like their siblings that stayed in the barn as the control group. No difference. Another angora rabbit myth debunked, the rabbits were fine.. They need dried hay (or dry grass, etc., uncut) to be available always so they can balance their intake and not founder themselves.Grasses and clover, alfalfa, & other legumes are particularly hot. I don't think most other forbs are quite as dangerous but it's always a good idea to introduce new foods slowly. And important to always have dried grass hay available so the animals can select what they need.
We've since retired from raising large livestock, but during those many decades we never fed hay to our horses/cows/sheep/goats during the growing season. They lived on pasture. Period. Grass-fed, no grain, no hay. Just like the wild critters. (We did occasionally offer grain, just to keep them coming when they were called--so funny to look over the hill and see them in the distance down by the creek, all spread out. Then you'd call--their ears would perk up, and suddenly the entire herd would be galloping up the hill for a treat.)
The problem is taking animals that do not have the correct digestive bacteria biomes, and totally changing their diet. Yes, it can cause problems. Yes, horses (and other critters) can get bloat from the early spring greens. We always kept their usual winter hay and grain mix out during the spring, to help alleviate some of that. You can also limit how much time they spend grazing at the beginning of the grass season. In some cases, the spring pasture was kept small, so there was less of that lush grass to graze, while the later pastures had time to grow before they were eaten down. (Grass that is still 3-6" tall after grazing will grow back in record time, grass grazed down to just an inch or so may take weeks and weeks to grow back, especially in a hot summer.)
But no, none of the farmers here feed hay to their stock during the growing season. It is an amazing thing to me, I've spent the last two years researching all of the rabbit "rules" of what you can and can't feed and do with rabbits--and in every case I've found people very very successfully breaking the "rules". (No, I'm not talking about feeding poisonous toxic feeds.) People that feed corn (gasp!) to their rabbits, national show winners that feed sweet feed with molasses and corn (double gasp!), people that successfully feed cabbage, or graze their rabbits outdoors, or have eliminated commercial pellets. A lot of this is due to the "breed to the feed" plan (I wish I knew who invented that saying). Some rabbits will do well with the feed you have available, some will not. Breed those that not only survive but thrive on what is available to you locally. That is how individual strains/breeds come to be--where people in an isolated region developed a line of animals that produced well on the feed/climate in their individual area.