I'm not sure what you mean by belief in actual statistics. Are you suggesting that they are the most important consideration in player evaluation, that their predictive value is the most important consideration in in-game decisions, or something else?
You mention older posters and today's advanced stats, but of course the work of Bill James has been around for a long time. And even Bill James himself cautions about relying too much on advanced stats to make decisions about players.
I agree advanced stats are a critical element of the game, but it's still a very human game for a lot of reasons, and why the game is played instead of determined by computer simulation. A good manager uses advanced stats, his knowledge of the game, his players, the opponent, etc. to win games.
It's also worth noting that studies show that luck influences MLB games at the rate of 28 percent. Advanced stats won't help here either. But wearing your lucky t-shirt will.