My favorite vegetarian lasagna to make is the Pampered Chef recipe (which I'm having trouble finding online) but it includes the veggies you listed plus carrots. I buzz all the veggies through the food processor (with the #3 blade) and saute them with a few herbs (oregano, basil & ???) then assemble. This recipe calls for precooked noodles. Oddly, no, the veggies aren't turned to mush with all that cooking. ETA: if I were using grated veggies, I wouldn't precook. But I encourage you to try the larger sliced veggies.
Other options are spinach lasagna, which I make with raw spinach & uncooked noodles + extra water & it all cooks together.