The stats and the eyes complement each other. Now on TV, the eyes lie a lot - you don't see the route an outfielder takes, even if the catch and end is pretty good. It is good when the numbers fit what your eyes see - Ozzie Smith's stardom was entirely justified analytically.
I think what WAR (which is a normalized sum of everything a player does that is measurable on the field) has advanced the conversation in some important ways. When I was a kid, you only thought in terms of pitchers and batters as far as getting down to runs. Yeah there were errors and wild pitches and such, but team defense as a meaningful contribution to run prevention was rarely talked about. We have a much better understanding (and it is still fuzzy granted) of the split between pitcher and fielder with regards to run prevention.