A Less Obvious Martingale Fallacy

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Assuming the game is not biased, and the wager is close to 50-50, Baccarat or the pass line for example, there does exist a number of series where the probability of failure approaches a theoretical zero.

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Doesn't your so-called "theoretical zero" correspond to a so-called "theoretically infinite dollar amount (loss)" on the other side of the coin?

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Assume the game was fair, and I could play a progression that lasted for 1000 trials -- I would be quite comfortable playing for a lifetime.

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Well maybe you should be. However, for all the people who might do it along with you and be successful and happy, wouldn't there be a few who would bear a commensurate loss to your summed profits by hitting that nasty old extreme negative end of the tail? (actually it would be more money because of the house edge on roulette or baccarat or whatever).