I think you are missing the trick here. Since HT and TH are equally likely, regardless the bias of the coin, as soon as you get one or the other, you can count it as a fair flip. All other combinations (HH and TT) are just ignored. The number of flips shown by the form is just about how patient you'll have to be to get a fair flip (HT or TH).

It doesn't matter how often you get TH and HT, only that they have the same frequency. Given then the probability of TH and HT are the same, if you only consider flips resulting in TH or HT, you automatically have a fair coin, no matter how bad (or good) the coin is.