I can't tell from what you're writing how much protein, then, do grass cuttings contain or how much a hen gets? That said, regardless, my main contribution - because this forum does exist for us all to provide our own experiences so that others might benefit from our experiments, is that my production blues and Eastereggers, whether they ate a ton of clippings or not, had the same egg output. You can't argue with what you see with your eyes. As for eating grass along with insects, anyone that's placed their hens in a protected/fenced area that was once part of a lawn knows that by the end of the summer, all the grass has been eaten. The ground is bare.
So you want me to do the math for you? On average fresh grass has a moisture content of 75-80%. If you put 10 grams of fresh clippings in a microwave and dry it, you'll end up with the dm weight, let's just assume 75% moisture was removed, the dm would be 2.5 grams. Of that 2.5 grams, we can assume 16% is crude protein, that is .4 grams of crude protein in 10 grams of grass clippings/that is 4% cp overall, and something in the range of 30% crude fiber. While crude protein is commonly used to estimate protein content, amino acids are a better measure of protein quality. I don't know the amino acid profile of your lawn grass, but I think it's likely a poor quality of the necessary amino acids for chickens otherwise commercial feed producers would just pellet grass. There is a reason you don't see grass fed chicken in the market, chickens do not possess the enzymes to digest cellulose, with nutrients locked behind cellulose, chicken get very little benefit from grass. Leave the grass to the ruminants.

Here in the forest we have moss covered ground, with bare spots wherever the chickens forage, they don't eat the moss, it's scratched out. Last year(edit: spring of '23) I planted a patch of Poa supina grass, the chickens occasionally eat it in the winter, (they prefer to eat the western hemlock needles) but it's very hardy grass, it's been run-over by the excavator and pecked and scatched by the chickens and it's still there. In Germany, the common name is Lägerrispe, which means, "where the cows lay". The name is a reflection of the ability of Poa supina to persist and even thrive on cattle trails, even in shaded woodland areas. If you want to continue to feed your chickens grass clippings, that's your prerogative, however, anything you add to their diet will have an affect on the overall crude protein and amino acid balance, crude fiber, crude fat, ME, etc. Do as you will with the advice you sought.
Just FYI: The studies that have been done on wild jungle fowl show that females eat far more vertebrates and invertebrates than do males, their reproductive system requires a higher level of animal protein.