Fortunately, they seem to enjoy picking at the tops popping through the grazing frame, but I've been curious about their interest in whole entire seedling / fodder matts
If I had the option of growing real grass or using a grazing frame where I live in northern Minnesota, I would do that instead of growing fodder. But our winters are long here and we don't have any grass growing outside for about 6 months out of the year. When I feed the fodder to my chickens in the winter, they eat everything including the root mat. In fact, some of my chickens prefer the mat, others eat the grass top.
Because my chicken run was eaten down to bare dirt, I started filling in the run with wood chips to prevent it from getting all muddy. Then I decided to turn it into a composting in place chicken run and started adding leaves, cut grass, etc... The chickens love digging in the compost and pulling out bugs and worms. The benefit I get is that the run is not muddy and does not smell bad. This fall I harvest about 50 cubic feet (about 50 bags) of chicken run compost that I added to my garden for next year.
I tried to use a grazing frame inside the chicken run, but the chickens would scratch all the leaves, wood chips, etc... on top of the grazing frame and nothing had much of a chance to grow. After a few weeks, I just gave up on the grazing frame idea. In my case, I mow the lawn with a bagger so I just give all the clippings to my chickens and let them eat what they want. They get more fresh grass then they can eat, but in my composting system, the uneaten grass gets mixed in with everything else and turns into compost.

From my experience this past summer, the more grass the chickens ate, the less commercial feed was consumed. My egg production fell throughout the summer as a result.
I am convinced that a well balanced commercial feed diet would probably result in a higher egg production. In my case, I get more eggs than we need anyway, so I decided to just let the chickens be chickens and watch them happily scratch and peck for their food in the chicken run compost. Having said that, I always have commercial feed available for them 24/7. They just prefer to forage most of the day for anything other than their commercial feed.
Point is, if egg production is your main goal, then feeding your birds natural grass and/or fodder should only be considered a supplement. I suspect that free range birds would also have lower egg production, but I can't let my birds free range due to predator concerns.