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> There is no convergence happening without the infinity.

This is wrong. It takes considerably fewer flips than "infinity" to see distinct convergence here.

If the coin is truly even - you have a binomial distribution, and the probability that you are far out to either edge of that distribution (nearly all heads, or nearly all tails) is vanishingly small with just 1000 flips.

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Probably, yes. But it's not a guarantee until you include infinity. And that's the important distinction here.

Without any such guarantee of the larger result, all you have is a 50/50 chance for the next flip. With infinity, you can pick out a portion of the sequence with an unusual number of Tails and expect there to be an offsetting number of Heads elsewhere in the full (infinite) sequence.

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