Cats, and dogs, need the green stuff, they really do. In a natural habitat, they would have seek it out. Here's the thing, spider plants are a delicacy for cats! They come in second only to the ponytail palm which is the absolute best treat in the world for cats. I buy both those plants for my cats to munch. I know, I know! They are so pretty and then they're mush. I gave up trying to keep one intact.
When they throw up the grass, most often what you will see is bile and a blade of grass at least an inch long,probably more. If I had that stuck in my throat, I would yuk it up. Have noticed that if my cats eat their greens before the meal, which they mostly do like us with a salad (there's no free-feeding here), then they do not throw up. But if they eat the grasses after the meal, they are likely to throw up a little.
All in all, grasses very good for cats. You can easily grow a pot or tub of yard grass. There's a mixture of rye, fescue & something else you can get from Home Depot, all of it safe for cats and they like it.
Now usually, I buy the small cheap ponytails when I can find them. This pic is one of the prettiest plants I've ever seen. It's a ponytail palm and the fronds are curly. Have never found another one. But I bought this knowing my cats would eat it and they devoured it in no time. I had to close them away to bring it in, they are that anxious for it. The pic speaks for itself.