"Effectively zero" times infinity is infinity. Infinity is like... really big.
Oh, I wish I could find it, but there's an excellent video by a mathematician explaining why an infinity doesn't work in your favor here. IIRC, it's because as the number of throws approach infinity, the odds also approach being infinitesimal, and you'll still lose.
But with enough flips, the odds getting heads on all of them comes close enough to zero that they are effectively zero.