I'd go further... the average number of direct reports is likely 7-8 people, but as you want to cluster direct report by team/squad to a manager sometimes they are unbalanced as the team/squad size isn't neatly divisible by that.

What you want to do is have the smaller teams/squads be the place where new managers can cut their teeth, and have the larger teams/squads be the home for managers looking to become directors. Assuming your statement about growth startup is correct the manager will eventually manage more people and the directors will have to deal with splitting the team, re-org and potentially then managing managers.

I'd add another thing to all of this: Why have managers?

The answer isn't the reason you think. It's not project planning, co-ordination, etc. Books like "Turn The Ship Around" or "The Effective Engineer" show how engineers can more fully own their role and not need micromanaging, and the benefits of that.

The reason for managers is different: Let's say that every 3 years a person has a life event. A life event is defined as something big, traumatic, emotional... both the ups and the downs: Birth, marriage, divorce, death, home move, home renovation, moving country, etc. In your startup of 10 people 3 such events will happen this year... and as a leader you can get involved and help support people. This leads to better retention, no loss of institutional knowledge, improved morale, improved loyalty (we're all human and when someone looks out for us we look out for them).

What happens when you have 100 people? ~30 such events this year. But each is large, will last many months and need a lot of support. Potentially you're now going to have 20 concurrent life events to manage.

At 500 people? ~160 such events to manage each year.

Managers and the People team aren't critical to your success, but their absence or a poor setup is absolutely going to be a large factor in the demise of your startup. The earlier you can structure the organisation to absorb these life events and really support people, the more likely none of this impacts your startup.

Then... once you have managers in place, great... there's a lot of spare capacity there to do more than just wait for the life events to occur. Now you can look at books like "An Elegant Puzzle - Systems of Engineering Management" and look to be a value-add to your team, supporting your engineers on a daily basis to be happier, healthier, more effective. Managers now can help nurture and grow senior engineers, help juniors find traction within their career, etc.