This is an interesting problem. It's tempting to boil it down to 'why RBIs are useless', but I think that that statement is both incorrect and overly inflammatory. So!
Prerequisites for understanding: None.
Prerequisites for derivation: N/A, conceptual.
Don't We Care About Runs?
The humble RBI is an easy statistic to love. After all, we've already been introduced to the idea of using linear weights and Base Runs in order to convert our statistics to runs. Surely making use of RBI, which is already in runs, makes a lot of sense, right? The idea of giving credit to runs scored to the player who drove them in is pretty compelling. After all, they're helping the team win. Clearly, people who don't care about runs batted in don't care about scoring runs. Or winning games.
To many, the RBI is the pinnacle of offensive statistics. Assigning credit to the guys who plated the runs seems like a pretty compelling argument, after all. However, this misses the point by a rather large mark, because while recording an RBI clearly means that a batter had something to do with a run scoring, it's rather unclear what that something was, nor how it compared to what everyone else in the play did. In other words, while we know there was a play and who the participants were, we have to be able to isolate the contributions of individuals.
The Isolation Problem
I call this the isolation problem, but there's probably a more official name floating around that I'm unaware of. In the case of RBI and runs scored, failing to properly isolate players from one another results in a fairly clunky statistic. Not that either one of them is useless, as is often stated - runs are clearly indicative of an ability to reach base, and having high RBI totals is a pretty good clue that a player is hitting well with some power - but they leave out the subtleties we'd like to see when assessing players. Should a player get full credit for an easy sac fly when the batter in front of him doubled the runner over to third? Should he get any credit at all if he gets a key triple with nobody on base and nobody drives him in? Obviously, in the case of runs/RBI we can reclaim a lot of information (i.e. credit) by looking at a player's individual statistics rather than the team-based ones, but there are more delicate situations to look at, too.
For example, what about the effect that hitters have on one another? Is there a protection effect at work? Does a batter behind a speedy stolen-bases type take more pitches than he would behind a lumbering power-hitter? How about fielders impinging on one another's ranges? Most of these questions have been studied already, and the answer has generally been that there is 'no' interactive effect ('no' because I believe there are effects, but they're too small to pick up on in these studies). While it's instructive to think about how players might be affecting one another on the diamond, a good rule of thumb is that if you have to wonder about whether there might possibly be an effect, it probably isn't important enough to worry about spending too much energy isolating. However, in obvious cases like RBI, the run-prevention abilities of the pitcher/defence unit, or even hitting in a very friendly stadium, we'll want to spend significant effort unpacking responsibility and assigning it correctly. This isn't to say that further work can't be done on the more intricate interaction between players, but our current measurements do pretty well, and we're just looking at marginal improvements in our metrics from here on out.
What follows
Park-adjustments; batting statistics; defence-independent pitching theory; defensive statistics.