I think when fans complain about "analytics" they're really complaining about one of the following:
-Management continuing to play a player who has been in a deep slump because the "numbers" say he's a good player. In the fans mind, "analytics" here means management having their noses in spreadsheets which compile data from previous months, or previous seasons, and ignoring the player's current performance.
-Management following a very simple top-level rule (righties should always hit against lefty pitchers, always pull the starting pitcher the third time through the order, hitters should always swing for the fences, never steal bases, never bunt...) and seemingly ignoring any complicating factor like which hitters are hot, whether the pitcher is cruising, whether there's a guy on third base and you only need one more run to walk it off, etc. What this fan is reacting against isn't "analytics," but bad, overly broad analytics. All those complicating factors are measurable, and could be used to make decisions mathematically.
-Management valuing a player for some esoteric stat (spin rate? launch angle?) and continuing to trust that statistic over the player's actual bad performance. I think fans sometimes even apply this to useful stats like OPS or WAR if the eye test tells them that a player strikes out too much or throws meatballs down the middle of the zone in clutch moments or whatever.
None of these errors are inherently what analytics are about, they're just a caricature of analytics.