That's probably easiest to see if you imagine approaching an infinitely biased coin (100% heads, 0% tails). Your strategy alternates between 0 and 1 almost always. The listed strategy throws away most flips but gives actualy unbiased results when a pair does pass.

Another way to look at it is from an entropy perspective. An unbiased, independent coin flip has 1 bit of entropy. A biased coin with, e.g., 99% heads has 0.0807 bits of entropy. On average, you need at least 12.377 such flips to emulate an unbiased, independent flip. Any strategy without some sort of rejection/continuation/... (like your proposal) is doomed to fail.

I haven't checked if their proposal is actually optimal. Empirically, it's suggestive of having room for improvement. I'm seeing something like 101 flips on average instead of 12.377 for that 99% bias example.