You have me confused a little because your original post seems to suggest you wanted it to equal 0:
> I was uncomfortable with the use of biases to assign non-zero probabilities to events that fail to occur after some number of trials.
Anyway, it entirely depends on your world view. When you say "There are other probabilities that can lead to zero successes" then that sounds like a Bayesian framework and that you have to pick your prior on the world. A natural choice is uniform (also Beta(1,1)) and update your prior as you collect data. You would then use the mean/median/mode of your posterior as your estimate for the bias. In your case, it appears you are operating in a Bayesian world but forcing your prior to be constantly uniform despite observing data. The frequentist perspective is that the probability is the relative frequency of observations. In this example, a frequentist would say that the probability of heads is 1 and tails is 0.
There are other probabilities that can lead to zero successes - every probability except for one.
And yes, they diminish rapidly as the number of trials goes up (much in the same way as it does in my equation).