WAR is the best we've got for what it does. It provides a (mostly) context-neutral way of comparing how many extra wins you should expect a player to have produced were he added to a random team. The values it assigns to events (singles, stolen bases, etc.) are based on the linear weight values of those events: basically, how many runs these events have historically generated, on average. (I don't know how far back the current weights go; Tango used, IIRC, five years when calculating wOBA in his book.)
You can quibble with parts of it. My biggest gripe is that it weights pitcher performance by the leverage index of the situation - basically it gives more weight to performance in close and late games than to things that pitchers do in the first inning. Consequently, IMHO, it overrates relief pitchers. And you can (and lots of people do) complain about how it handles defensive statistics (especially since it uses different measures of defense for early players than for more recent ones). But these are quibbles, not objections to the WAR framework.
fWAR and bWAR are different stats which measure (slightly) different things. To object to WAR as such because there are two versions is like objecting to batting average because on-base percentage measures a lot of the same things, i.e., it's a criticism that doesn't make a lot of sense. (Although batting average is objectionable on other grounds.)
As with any statistic, the important thing is to not misuse it. That Barry Larkin was worth 6.1 WAR in 1991 (which is very good, all-star quality play) doesn't mean that the Orioles should have traded for him, since the Orioles already had a pretty good SS themselves. So that would be a way of misusing it.
The problematic stats, like saves and pitcher wins, are problematic because either there are very few situations in which they're useful, or because they're so consistently misused. There's a time and a place for citing wins. For example, if you've managed to win 300 games, you're going to be a pretty good pitcher. Not because wins are a good way of measuring pitcher quality (they're not), but because bad pitchers don't stick around long enough to win 300 games. So if you didn't know anything else about Early Wynn, pointing out that he won 300 games is a good way of pointing out that he was a really good pitcher. But they're not useful for much beyond that. (Because a pitcher's team mates make such a big difference to whether or not he's going to win any particular game.)
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