They really aren't.
Sure, some of the advanced metrics are not as simple and clear-cut as 'this stat shows this %'. But they all incorporate the same basic data, the advanced metrics just do so in a much more in-depth way to attempt to explain things better. You just see the final product, which is 'this player is worth X WAR'. You don't see all the underlying data, or statistics, but that's all it is: a calculation of statistics.
So while BA will always tell you exactly what % of ABs end in H, and in that sense be completely accurate at all times, and WAR is more complex and may not tell you literally the number of wins a player was worth or contributed, that doesn't necessarily make BA a better measure of anything. The issue with BA is what it doesn't tell you and the fact that it is a very poor measure of a player's overall worth. The issue with WAR is that it isn't clearly measured and may not be entirely accurate. What I don't understand is using one as some kind of absolute measure of worth while discounting the other entirely. IMO, BA should be used for exactly what it measures, and WAR should be taken into account as a rough measure of a player's overall ability and taken with a grain of salt. They are both using data to describe something, and both are useful if used correctly.
In the end, though, statistics are just particular interpretations of data.
I've never heard anybody argue against data in and of itself (except, perhaps, quibbles with collection methods), but I've read many criticisms of statistical models.
I mean ... the data is the game itself.