Add in a specific seed for grass. This grass would have the ability to be planted on plowed land like other crops, but it will have a different way to increase in quality.
This grass will be treated like a crop, but it will really be just for harvesting grass. For hay. So we can stop feeding animals that eat grass carrots and beets.
You can harvest large swathes of it at a time with the scythe, its ready to harvest every two or three days, and should have a recipe to build it into a hay bale/bundle. Or really big stockpiles that can go into carts.
If the bale route is taken, it could be rolled by the player at crawl speed, and provide food to animals with a smaller radius than a through.
I was thinking that this grass would be something you plant once, and never pull it up. A little like how trelli work, but the grass will have a unique way of increasing quality. This way you can harvest the grass when its ready as a food source for animals. Grass's quality would need to be increased by compost, or manure if that is added. There would have to be a time cap on how often this can be done. With this system, you will still have to work to increase the quality of your grass, but you won't have to deal with replanting it, or ripping it out for the new quality seeds. The amount the grass can increase in quality which each fertilizing should be capped at 1-3 depending on the quality of the fertilizing material vs. the grass's current quality.
As for harvesting seeds, I dunno, just let the grass drop some seeds upon harvest. I was thinking initially that it could have two stages where grass could be harvested, but the second one could also provide seeds. Seeds will always be the quality of the current grass, not higher in quality. People don't grow grass by ripping it out every time they mow it or harvest it for hay. They fertilize it to keep it fertile and growing quickly.
Grass seeds would only be for the purpose of planting more grass or feeding to chickens.
Note: For this idea, the grass will only grow like a crop if its planted on tilled ground. Its seeds could still be used to spread normal grass like other seeds today.
If seasons are ever added, making grass grow 3x as slow in winter would be a nice addition. This would encourage stocking up on bales of hay for animals, and encourage trade for those who didn't store enough to get through the cold season.