While an interesting idea, not much of a way to prove it(as you say).

I look at it another way.   The difference between a .300 hitter and a .250 hitter, based on 600 AB, is one hit every 3ish games.    Roll the dice 600 times and you'd expect 100 for each number.   That won't happen.   In all likelihood, you'll get a each number between 80-120 times.   Weight the dice and you increase the chances of getting specific numbers more often but you still won't guarantee it.  

HBD players are weighted dice on a slanted table.   Some results are more likely but not guaranteed. 

I agree that finding a smoking gun will be next to impossible to find.  Still I would suggest the math actually points at some manipulation since flipping a coin 600 times 10 seasons in a row has a way of normalizing the results.  

I think that there is in fact some form of modifier and that it has a lot to do with the team roster as well. Players play better on winners and worst on losers. It works like that in real life and it makes since in the game . Some of the swings you see in stats are not other wise justifiable. 330 hitter now hitting 275 or pitchers who swing from a 3.50 era to a 4.50 all the while in the same ball park with the same fielders from 1 season to the next.

I think winners win because they play well, and losers lose because they don't. I would love to see the database and the code that runs this stuff but I suspect it's much simpler than many of us think.

I, too, like the 'secret sauce' idea.  However, there's a couple immediate factors that can readily explain a lot of season-to-season variation:

1. Competition: did a player change to a weak division, where they face relatively weak opposing players?  Since the talent distribution in a world is far from uniform, players can have brilliant careers in part by spending 10 seasons in a backwater division.  

2. Settings and management: very simple team-to-team variable.  Example: a guy who can steal 2B 90% of the time gets stuck on a team with 'conservative' or 'very conservative' base stealing settings.  He steals 20 bases.  Put him on an 'aggressive' setting and he steals 60+.  

If there's a variable modifier for player performance, I suspect it's a hot streak-slump trend.  It would be easy to bias players toward one or the other using the ratings already in the player profile, such as patience, temper and makeup.  Does poor team or individual performance beget more poor performance, or does a player's makeup help them overcome their doldrums? 

Would be super-easy to program the game this way, and I would have done it.  Every week, reflect on the previous 2 weeks performance, roll the dice and see if a player gets complacent or gets sloppy or fixes his swing or eliminates a tell in delivering his change up.  

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Separately, if the programmers ever read this stuff, I think a good rating to add would be 'idiot factor.'  It's the only rating that goes DOWN in a good way.  Kids do stupid things, vets do proportionally fewer idiot things.  Puts a small premium on veteran status, so even as raw physicality diminishes, some degree of savvy can partly make up for it.  

Like Mike said "HBD players are weighted dice on a slanted table. Some results are more likely but not guaranteed."

Is 100 percent correct, over a season best teams will have best records. However in a small sample size of a playoff series the best team will not always win. Likely to but not always... There is a certain randomness or modifiers that come into play but exact recipe is only know. From
Coders...

I like the vet idea or also a clutch style of rating... Like people
You want taking big at bats or people you don't and would pinch hit for!! Same same for pictures...

I haven't had a lot of time to check more than a handful of players but, using 10 seasons on my team in the same ballpark as a guideline, I found the following:
Pitchers are remarkably consistent.
Position players are reasonably consistent but are far more likely to have an outlier season(good or bad).

I think this can be attributed to a multitude of variables.  First, I build my defenses pretty much the same so my pitchers get consistent "help".   Second, the opponents are going to be somewhat similar from year to year.   You'll be hardpressed to find a world that has an average OPS of .720 in one season and .750 the next.   I believe these are the main factors why my pitchers consistently crank out similar stats from season to season.   Hitters are different for pretty much the same reason.   Upgrading defense is pretty easy.  It can be as easy as replacing your SS and rotating everyone else around to a position better suited to their skillset.   As far as pitching they face, they could get in the rotation of facing 3/4/5 starters in most series as opposed to 1/2/3 guys.   The difference between the 1/2 pitchers and the 4/5 pitchers are likely huge.

because the game engine itself is based off of either decision-trees or some type of single algorithm (or both) which produce thousands and thousands of randomized independent trials, there can be a substantial luck element involved sometimes...

The only thing I tend to ever notice to this point is that rookies seem to under-perform more than their ratings themselves suggest they should (as examples, I understand it's s.s.s. but Hardball Dynasty – Fantasy Baseball Sim Games - Player Profile: Nick Holmes  Hardball Dynasty – Fantasy Baseball Sim Games - Player Profile: Jesus Miro tend to both "lose the algorithm" constantly as if they were fringe players rather than rookie-of-the-year candidates, as their ratings suggest)

Also, in reference to your Charlie Hustle point, if you consider a player like  Hardball Dynasty – Fantasy Baseball Sim Games - Player Profile: Kent Ward, Simmy actually tends to cross-reference different attributes when it spins the algorithm.  Generally, batters are programmed to utilize their eye, contact, power, and splits ratings (I think sequentially in that order actually also), but because I read thru the game summaries line-by-line, I've noticed that for this particular player, he is mostly only subject to contact, bunting, opposite field, speed, and baserunning (I think sequentially in that order as well).  This player appears to only be subject to splits and power in situations when he's actually swinging the bat (multiple runners on base, etc).  In those situations, he is mostly so-so as it pertains to walks, home runs, and base hits.  His 100 speed rating compensates for his poor splits because he beats out a lot of infield singles, and therefore his batting average is way higher than it would otherwise be if he was a player with ordinary (or merely above-average) speed.

In summary, it's important to remember that there are literally thousands and thousands of independent trials per player per season, so variance is to be expected. Also, it's worthwhile to consider the sequence in which the algorithm actually spins, and the degree to which manager settings pick and choose the attributes that are getting spun