Non-rated games already don't have any impact on players' ratings, which is as it should be. Likewise, they should have no impact on W-L-D records.
What if the current #2 player wanted to coach some players here? Perhaps he would want to play some practice games against them--maybe giving them odds, or setting up endgame scenarios, etc., games that he was "supposed" to lose to the trainee. He might be discouraged from doing that if he likes seeing that he has 0 wins.
Perhaps he goes ahead and loses 30 unrated games. He doesn't care about his record, just his rating. But others on the site observe his record and say, "He's got a great rating, but he's beatable, and he's a lot worse than some lower-rated players, who have a lot fewer losses."
Including non-rated games in W-L-D records distorts the various presentations of statistics, confuses people who seek information from the records, discourages teaching and practice, and punishes people who might want to play "friendly" games that might be used to attract people to the site or to chess in general.