It's not the "more variables" put into the process, it's the more components that make it better than using any other single stat or most other metrics.
It's an attempt to quantify the value of offense, defense, base-running and positional differences into one number that gives comparative value between players.
It's easier than saying player A is better than player B, because he has a higher OBP and more HRs, but is worse on defense, running and slugging. Talk about confusing. The "old way" of comparing players has always been subjective. WAR tries to objectify the subjective. It's not perfect. I disagree with some of their positional values, but at least they have data and research to support why they value each component a certain way compared to others.
The end result is that when I look at the top 10 or 20 lists by WAR, they look pretty damn close to what I think the order should be. It's closer than ordering players by OPS, wRC+, UZR150, HRs, BA, etc... One thing WAR does not do is to credit a great player who has missed time due to injury or lack of playing time. Player A may be "better than" player B, but was out hurt for a while. WAR just captures the value of what you have given in a certain time period.
To me, WAR is just a tool to try and reduce the whole of a player's skill level to one number. It's pretty darn accurate, but it's not the be-all-end-all and can be improved upon. There's still room for argument and discussion using old school stats and observations, but WAR gives a very nice guideline or framework for any discussion.
To me, the onus is on the person claiming player A is better than player B despite WAR showing otherwise.