[SplawnDarts]

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Averaging is only useful if the mean of the things you are averaging equals the accurate value you're trying to get. In this case, the value we're trying to get is our expectation with respect to villain's hand.

There's absoloutly no rigorous reason to believe that the expectation for a number of hands averages to the expectation for the hand villain actually has.

So in fact there's a subtle problem with the process above and beyond any problems of execution.

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This looks pretty dodgy to me. Would you care to give an answer to a simple thought experiment, just to see if we are on the same page?

Suppose you are given an opportunity to choose either

A) a guaranteed payoff of $100

or

B) a chance to choose from one of three boxes, with each box having a different amount of money inside it. Let's say the amounts are $50, $80, and $200.

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The way you described it, B).

However, your situation B) is not analogous to anything that occurs in poker except for a blind all-in. Do you see why?