But also, you might have to flip the coin an arbitrarily large number of times before you get a "heads tails" or "tails heads" roll (if I can arbitrarily pick how biased the coin is).

The opening scene of "Rosencrantz and Guildenstern Are Dead" springs to mind.

And that coin wasn't even biased... although Tom Stoppard was a confounding factor.

You are assuming an unbiased coin.

Imagine I glue a poker chip to a washer. There's a clear bias in the outcome of this "coin".

This method resolves that bias.

I understood perfectly already.