Yeah, the one free relief per round idea seems like the only realistic option. You can't make it a judgment call, and playing it down is fundamental to the game of golf. I'd also add you can't clean it and you can only move it a tee length or a putter head or something tiny. A lot of courses give relief for sprinklers in the way of a putt from the fringe now, I feel like that's a similar accommodation - it's just easier to identify a sprinkler head as a manmade object versus a divot.
Another reason I'd be ok with it despite being fairly traditional when it comes to golf is that divots are one of those issues that tend to impact amateurs more than pros. As has been pointed out, most pros land in a sand-filled divot on an immaculately conditioned course, and 9 times out of 10 they hit a decent shot. If you play average golf courses with the typical idiot public, you get the occasional horrible 2 inch ditch with zero sand, or you get overplayed narrow fairways that are covered in divots. There's a hyper exclusive country club near me that has a local rule on a couple holes that you get relief from divots just because the fairways funnel balls to a general area and it's an uphill approach, meaning that if you roll into a divot that hasn't been filled you're basically losing a full shot. For USGA events they play it down and it sucks, feels like you have a 25% chance of getting screwed and guys definitely aim for the rough because it's such a risk. An official, even optional rule of one slight move per round without cleaning seems fair enough, plus it's not really possible to take advantage of it.
The underlying theme is that the rules of golf consistently allow relief for man-made objects and impediments. They also now allow for the repair of any and all blemishes on greens made be other players. The only reason divots aren't covered is the difficulty in assessing them paired with the infrequency of the issue occurring. The "one divot relief per round" rule is a pretty elegant fix, and imho would even encourage more players to play by the rules in general instead of using lcp all the time with friends.