But you typically don't get to be a director with an org of 5-10 people, at least not in engineering, which OP indicated he's in. (It's different in other functions. PM/UX directors frequently have 10-15 people under them, and my wife is a deputy director in finance and has 2 reports. Her boss is the director and has 3 indirects, just her and her reports. I joke that their management chain really is a chain. Startups are also different - I once interviewed for a VP role at a startup that had 8 employees.)

For that promotion to director, you usually need at least 50 reports and at least 2 levels of management. And then the span of responsibility for (senior) director goes on up to about 500 reports. It's not unusual to have directors with less than that - but usually that is because they once had a big org, their scope and responsibility decreased in some re-org, but upper management is keeping them around so they have a deep leadership bench. Another re-org and they can easily end up with 500+ people again.