OK, I've got it: why not manufacture synthetic turf out of cornstarch or some other biodegradable polymer, like the packing peanuts?
Obviously you don't want it to degrade in a day, but couldn't the lifespan be about a year if just enough insoluble fiber or milk proteins or something is added? Or some marginally stable organic compound that doesn't decompose in the presence of water, but rather hydrogen peroxide? And for that matter, couldn't you imbue it with microbiology to keep some semblance of soil food web going down beneath the drain rock?
Boy, the syn-turf companies are really missing the boat on this one. Annual maintenance or infill replacement be damned -- just sign up for the scheduled annual replacement! And they wouldn't have to mess with all those 5- and 10-year warranties... I can see the marketing now: "place your order now to have your brand-new BioLawn(TM) delivered just in time for spring training!" Installation would be a cinch, just staple the new one down on top of the old.
Seriously, is plastic really the only thing we can make grass out of?