You are perfectly correct. A simple distillation is rarely sufficient to purify a compound from a mixture, unless one of the components of the mixture is non-volatile. For example, Discodermolide's salt-water example.
Look at a phase diagram of ethanol and water - for example,
http://www.chemguide.co.uk/physical/phaseeqia/bpcompn1.gif. Let's assume that you are starting with a mixture that is 1:1, right in the middle of the diagram. You start to heat the pot, and the temperature rises. What happens?
Draw a vertical line from the 50% by mass point and note where it crosses the liquid composition line. This is where your solution begins to boil. The vapors that are being collected above that liquid, however, do not have the same composition as the liquid. If you draw a horizontal line now from where your vertical line crossed the liquid composition line until it crosses the vapor composition line, you will find the vapors above that liquid are enriched in ethanol. This is the composition of the vapors that you are condensing. As you noted, there is still water in this mixture, but the mixture is enriched in ethanol. You could take this mixture and distill it further to generate purer ethanol, and keep going until you reach the azeotrope.
In practice, this is carried out by fractional distillation. In theory, you are setting up a whole series of boiling pots, starting with one at the boiling point of water and ending with one at the boiling point of your azeotrope. You do this by setting up a long column over your pot, and maintaining a temperature differential along the column. The bottom of the column is at 100° and the top at 78°. When you heat up your pot, pure water will condense at the very bottom of the column, and drip back into the pot to be boiled again. Slightly higher than that, 10% ethanol in water will condense, and as it starts to drip down the column, it will get hot enough to boil again, sending the ethanol further up the column and the water further down.
Here is a page that describes the process with some better visual aids...
http://www.chemguide.co.uk/physical/phaseeqia/idealfract.htmlHope this helps