> 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.

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.

I found myself struggling with the intuition of this similarly to amingilani until I made the connection you're mentioning to infinity. The thing that helped me understand the fallacy was understanding how truly large infinity is. If you've seen 10 coin flips, or even 10,000 or 10,000,000, those are all effectively nothing when compared against infinity. If you know ahead of time that a coin is fair (so you're not trying to test this hypothesis), you gain zero information about the long-term behavior of the coin from any finite number of flips.

As you mentioned, you don't see the convergence to 50% with any finite sample size, because that behavior is only guaranteed at infinity. All this to say, infinities are weird. 10,000,000 + infinity exactly equals infinity.