I think arbitrary is the wrong word. The LSAT tests how well you can read dense passages and sort what's important from what's fluff. It tests deductive reasoning abilities. It tests you on your ability to discern a good argument from a flawed one. And so on. These are all skills that, at least to me, seem important during your law school education and in practice as well.
In terms of fairness, I don't see how it isn't fair. It provides a way to put everyone on a level playing field. What's not fair is that the majority of schools don't give a shit what your major was. That a 4.0 in a bachelor of breathing and walking is considered more competitive than say a 3.7 in a rigorous degree like engineering. And logically, that makes sense. Figuring out some sort of weighted average to compare differing GPAs from different programs seems highly unrealistic. But it's still bullshit. Same goes for clubs. I myself was extremely fortunate in that I had the means and the time to join clubs and whatnot, so my "softs" were quite excellent. But people in other circumstances simply may not have time for clubs. Maybe they're too busy working when they're not in class, or they had to take care of a sick family member.
All in all, there's clearly evidence that the LSAT does correlate, at least somewhat, to success in law school. And more importantly, it's a quantitative factor, which in and of itself means that it can more easily be used to compare 2 candidates. It gives the adcom an additional hard number upon which to compare candidates, and in my opinion, that seems more fair and accurate than assessing applicants without it.