The pellets do have extra nutrients including a type of Vit D, which is important especially for indoor bunnies who may not get enough sunlight. As far as the T part (Timothy), it’s higher in fiber and lower in calorie than many other based pellets that may be made from alfalfa (which is okay young bunnies under 7 months)
Some people do go pellet free and just feed hay and greens. I actually have to do that with my bunny Jack due to specific health issues he’s dealing with, but it does mean i have to feed him other things like a small amount sweet potato and carrots as part of his daily diet (not just a treat). Normally these things would be too high in carbs/starches in addition to pellets and can cause digestive problems.
As far as Timothy Hay – you can also use Orchard hay, and mix in other hays, but Timothy is known as the basic for the for the same reason – high fiber and lower in calorie – the 1st cut Timothy hay is higher in fiber (it’s courser). You do want to stick with grass hays for adult bunnies, and not use legume hays like alfalfa and clover because legume hays are higher in calories and calcium (some bunnies have a harder time with that) unless it’s for young bunnies or if you are just giving the hay as a very small percentage as a treat or to encourage hay consumption/. (unless your bunny is already overweight, has soft stools, and other digestive upsets)
Dandelion can be included in the veggies (veggies should include dark leafy greens) Please see: https://binkybunny.com/Default.aspx?tabid=144 for a list of greens/veggies.
As far as branches, you do have to be careful about what type. Apple branches and willow are commonly used and are fine. Though if you are pulling stuff just out of your yard, you have to be very careful about making sure the sap is washed off, and dried properly. I know blackberry leaves have been discussed, but I really don’t know about the branches. So that is something I will have to research unless someone else knows the deal with that.