As a tangential question, how often did that happen? In the movies, a final push to overtake the castle is very dramatic to watch. But how often did that happen versus, say, just starving out the people inside?

Similar to the original question, I always wondered why the defenders never just piled rubble behind the gate. It seems like no one would be able to break down your gate (even with a ram) when there are severals tons of rock behind it.

I was always under the assumption that you would starve/disease a city over several weeks/months/years until they were so weak that you could storm the tiny garrison or force them to let you in. Could someone clarify this?

It was not uncommon – especially in the high and later middle ages – for castle gatehouses to be designed so that they were almost a fortress within a fortress and able to be defended separately against attackers who were inside the enclosure as well as those outside. There would be doors (and sometimes portculli) at each end of the gate passage, entrances to the gatehouse from the interior would be strongly defended, drawbridge and portcullis mechanisms could only be accessed from within the gatehouse etc. City gatehouses were often similar.

Harlech Castle in North Wales has a classic example of this kind of gatehouse.