The Beta distribution with parameters $a$ and $b$ can be thought as the posterior distribution of the probability of heads when we start with a flat prior and observe $a-1$ heads and $b-1$ tails. And the parameters $a$ and $b$ can both be anything from $(0, \infty)$.
If we reparametrize it via: $a'=a-1$ and $b'=b-1$, we end up with a distribution where the parameters represent the number of heads and number of tails instead of having the $-1$'s tagged on. The only catch I can see here is that the parameters of this distribution can lie anywhere between $(-1, \infty)$ rather than $(0,\infty)$. But this seems a small price to pay for the more natural interpretation.
Am I missing anything? Is there any other reason we don't reparametrize the Beta so that its parameters represent the actual number of heads and tails observed?