Why is the beta distribution so flat when a, b=1? If the beta distribution is a prior of a Bernoulli distribution (i.e. a rate of success for a binary outcome), then it is completely counterintuitive to me that the beta distribution should be equivalent to the uniform distribution when a, b = 1.
Unless I'm mistaken, you can interpret a and b to be the number of measured successes and failures of your outcome (e.g. heads or tails of a flipped coin). If that's true, then the likelihood of your posterior Bernoulli parameter being either 0 or 1 should be much closer to 0 (i.e. and NOT equally likely as values closer to 0.5).
If the underlying Bernoulli really was as likely to be 0 or 1 as something closer to 0.5, you would NOT expect to get both a 0 and a 1 out of two samples, right?
What am I missing that would make this more intuitive?

Every conjugate prior distribution to an exponential family has some set of parameters $\eta_0$ that result in a uniform distribution over the space.  You can see this here by considering what happens to the natural parameters $\eta'$ of the conjugate prior when the number of pseudo-observations $n$ equals $0$.

Unless I'm mistaken, you can interpret a and b to be the number of
measured successes and failures of your outcome (e.g. heads or tails
of a flipped coin). If that's true, then the likelihood of your
posterior Bernoulli parameter being either 0 or 1 should be much
closer to 0 (i.e. and NOT equally likely as values closer to 0.5).

That is true, but beta$(1, 1)$ corresponds to $a=b=0$.  The natural parametrization of the beta distribution is $(\alpha-1, \beta-1)$ compared the "source parametrization" (or common parametrization) of $(\alpha, \beta)$.
