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There is a possible bit of equivocation in the term "evidence".  On the one hand, evidence could be understood as any data point that would count in favor of a hypothesis, in which case, there is no doubt lots of evidence for a protected hitter effect.  In fact, each hit by a hitter preceding a big bopper is "evidence" in this thin sense.  However, this can't be what the sabermetricians mean.  Rather, the claim "there is no evidence that P", if interpreted charitably, should be read as "given the available evidence, P is clearly not justified".  In fact, in common usage, this is what people usually mean when they talk about there being (or not being) evidence.  Much of what you've written seems like lack of charity on your part, and a failure to grasp a possible equivocation in the term "evidence".
 
A  few questions for Doc:
Is it possible for someone to believe  that P is evidence for Q when in fact P is not evidence for Q?  
Is it possible for a person to believe that some set of beliefs X are sufficient to justify a further belief Z when in fact X is not sufficient to justify Z?
 
I'm just wondering how much of a relativist you are about issues of evidence and justification.