So I think what you heard is about the right idea. The flat region is your buffering region, and isn't super helpful to deduce the pKa. Adding base consumes the weak acid, and because it is weak, you know you are mostly consuming the [HA] form, and equilibrating back to around the pKa. This is why your pH doesn't change much around the pKa.
As an example, if you had 10 mmol of HA to start, then at the pKa point you would have 5 mmol of HA and 5 mmol of A-. Now, if you keep adding base, at some point you will essentially consume the remaining 5 mmol of HA. Then, there will be negligible amount left and it can't buffer anymore - i.e. your pH will change rapidly because you are adding strong base. Here you will have ~0 mmol of HA, and ~10 mmol of A-. Note that you have twice the amount of A- now. The pKa will have been at the point where you had half of this.
Experimentally, I know two simple ways. Using a pH meter and no indicator, you have to measure the sharp region more carefully (i.e., drop by drop), because you want to be able to find the exact point where the pH changes the fastest. By approximating the derivative, i.e.
$$ \frac{d}{d(V_{base})} pH(x_i) \approx \frac{pH(x_{i+1})-pH(x_{i-1})}{x_{i+1}-x_{i-1}}$$
you can find the point of greatest change, or the equivalence point. Obviously, better data in that region will give you more clarity. I used to do one titration quickly so that I would know the approximate region where I would need to be and then slowly titrate the rest.
The other method, i.e. using an indicator solution and then just very carefully reaching the equivalence point essentially gives you the same information, and I know people who can very quickly do this. But in general, this would not be as accurate.
