While I don't claim to be the math expert here, each of those flips were at separate events with different people calling the toss. None of them affected the other, each was a single event.
It's the same fallacy as thinking because the roulette wheel came up black 6 times in a row it is due to fall on red. No, it isn't.
This is exactly why you can use the geometric distribution and get a 1/64 odds - each is a so-called independent Bernoulli trial. You can also use the Binomial distribution to find the odds of any particular amount of each outcome for a set of 6 coin tosses; you will find that 3 is the expected value.
Here is why: consider using 6 different coins for clarity's sake. You like them up in a set order and write out the outcome, e.g. HTHHTT.
Now, by combinatorics, there are 2^6 = 64 possible combinations. But not all of these represent different outcomes in terms of number of heads. For example, HHHHHT and HTHHHH both give the outcome "5 heads, 1 tails."
The probability of any outcome then, is the number of unique combinations representing that outcome, over the total number of unique combinations (the binomial distribution calculates exactly this.) For example, for the above 5 Heads, 1 Tails outcome, we may represent it with six different combinations (the single tails can occur in any one of six places), and so the probability of that is 6/64 or 3/32 ~ 1/11.
For all heads, or all tails however, we are in the unique position of having only one possible combination representing each: HHHHHH and TTTTTT. Thus, either of those has only a 1/64 chance of occurring. By comparison, there are 32 different ways to get 3 heads, 3 tails.
I hope this is clear enough. This is really just an elaboration on the principle of multiplication, a basic statistical law easily derived through the definition of independent events.