The convergence appears pretty quickly.
This is your faulty assumption. It does apear pretty quickly. In most cases. But not at all every time.
There are in some sense two layers of likelyhood: In layer one, every single event has the very same probability as its precedessors. In layer two, the sequence of events as a whole has a probability to occur. And every single sequence of a given length has the very same probability to occur, ie. HTHTHTHT has the same probability as HHHHHHHH (H=heads, T=tails), which is 0.5^8. It is only because across all possible sequences of a given length the number of heads and tails is 50% each that generally, a sequence of independent throws converges towards these frequencies. And of course, there are many, many more series of the length of eight tosses which contain at least one tails, which makes us think that there will occur tails pretty soon.
The problem is that you never know which sequence you are in, as it were. That is why, looking to the future, only the probability of the single next event is what should count for the gambler. The improbability of the 11th heads after 10 times heads is purely subjective because there are so many more sequences with at least one tails, but it is still 50%. After all, having 10 heads in a row is the improbable event, not that the next throw will be heads again. But, well, it still has happened, so it makes no difference as for the next throw.
You have to see what exactly the event (and object of probability) is. In the coin-example, the series so far is an event that has occured. So there was a probability for this series to occur before, but now, as it has happened, there is only a frequency of occurence left which we already know. The only probability in the strict sense of the word, which is about predicting future or at least unknown states of affairs is that of the next event or of the upcoming series. As soon as the next toss is made and we have seen the result, there is only the probability of the now next event and the now following possible series.
The fallacy consists in assuming that because there are ever more possible sequences with at least one tails the longer the overall sequence gets the probability of tails after him experiencing a huge number of heads would increase. But no matter how improbable the sequence he already encountered was, it is a category error to apply probability to past and known events, ie. "his" sequence +1 toss compared to all other possible sequences of this length. For each toss, the probability is still 50%, no matter what happened before.
Probability proper does only make sense for/applies to future or unknown matters of facts!
As a sidenote the following "counter-example": Consider three doors, you choose one that has the "probability" of containing the prize of 1/3. Now one door is opened and you have the choice to change the chosen door. What are the chances? Well, you should definitely change, because your door now has the "probability" of 1/3 as before, but the other has 2/3. Here you have to consider the whole series, there is no contradiction. That is because there is no probability anymore: The prize already is behind one door, the event has happened. That is the difference.
TL;DR: Edit and Conclusion
So the fallacy, as expressed by @wedstrom in his comment, is to think that nature will correct itself, will let it happen that the series in progress will converge quickly. But nature is not an actor that does anything. And in the present, there is only past (occured/known events, frequency) and future (upcoming/unknown events, probability). Thus, if the probability is independent, this has to be taken literally as independence from anything that happened in the past, no matter how scarce the occurence of the resulting overall series does appear.