In my 4E hack, I have a rule that says:
REMAIN IMPARTIAL
Your job as DM is to remain impartial to the success or failure of any PC or NPC. You do not pick sides. You have no preferred outcomes. You do not determine any outcomes before or during play; the actions of the characters in the game and the result of dice rolls will do this for you.
The DM can't break this rule without cheating.
A "true neutral" GM will never happen due to the fact that GMs are humans, not machines. Maybe you're VERY good at it, but I'm willing to wager when there are moments when you're partial to your giant monster and smile as he bashes your PC's heads in. Likewise I'm betting there are moments where a good play by a player will make you chuckle.
Impartiality to the numbers? Very doable. But the goal of every game is to have fun, as such, your impartiality should be, at best, utilitarian. Be as impartial as possible, until you realize that just because the game says that X should happen because Y just happened, it's going to make for a very bad gameplay experience.
ie: In WoW on a pvp server, you WILL get griefed. Because they rules say that they can and there's nothing you can do about it because you're too low level. To take this to D&D, an arch-demon from the 9th circle of hell can show up and massacre your lvl2 party because they happened to kill a young succubus he was about to get it on with. This however, is dumb and not fun. Therefore, most high-powered monsters show up, do something flashy, taunt the players, laugh and then run away. Which is generally considered to be fun.
So yes, I agree you should stay impartial to the numbers, but only as long as you are not remaining impartial to goal of the game, which is to maximize fun.
The goal of the games you're comparing are quite different. In sports like baseball or football, the goal of the game is to play to win, not just have fun or have a shared experience. The rules are, therefore, much more important to provide a fair game so that the win is determined by pitting skill vs skill, not skill vs superior ability to cheat the rules.
Originally, "winning" was fun, sports were more friendly competition and less hardcore money-grabs.
The goal of a role playing game isn't to win. Aside from various tournaments, there's no competition between players that will determine "a winner". There's no need for as much strict adherence to hard rules to provide fairness unblemished by a player gaining an advantage or two.
Well, to an extent the goal is to win. The players are trying to overcome the challenges, and the DM is trying to overcome the players with the challenges. But it's a much friendlier competition, and yes, competition can, and IMO should exist between players. ie: the Gimli-Legolas orc-killing competition.
There's also the nature of the game. The nature of most sports is determined by what the rules define the sport to be. How do you tell who wins in baseball? The rules tell you exactly how you can score.
In RPGs, the players take on the roles of avatars interaction with a fictional world but real from the avatar's perspective where they can do pretty much anything they want within their defined powers and free wills. They're not constrained within a certain playing area, to particular game durations or innings, to limited forms of contact with other game participants. The rules have to be treated as incomplete and open-ended. They may not be able to completely define how an avatar's action would play out, hence, the flexibility a player or GM has to determine the downstream events that unfold from a PC action, overruling written rules as necessary to do it (and the accolades they get if they do a clever job at it).
I agree, I think generally the group(DM included) is victorious in the game provided that everyone had a good time.