There's (one) of your problems. Your CTO (just like managers) should not be coding, they should be defining and leading the technical strategy of the company.
It's a joke to give yourself a management title when there's nothing to manage.
Being a 2 people company or a 20k+, you still have to manage financial, technical, operation aspects of your business.
Same role name, different implications depending on scale.
If you are a small handful of people starting off, you don’t need a CFO, you need a bookkeeper and an accountant. You don’t need a CTO, you need a tech lead.
All that rest is a mix of confusion and puffery.
Ignoring that when giving yourself a C-level title is what I take issue with.
Same role name, different situational implications.
Being a mayor of a small village or of Paris or New-York, you're still the mayor.
So really there's no practical difference between a company of 11 people and a company of 11000 people from a management perspective.
The technological/practical concerns you'll be dealing with are totally different, of course, but as far as 'people management' stuff goes, there's little difference.
In fact, it's not even with the people you're directly dealing with where this becomes useful (as, if you're selling to Big Enterprise Client, you likely have a network inside the company anyhow). It's when the person you have a relationship with has to convince his/her boss to approve an invoice from your little shop.
At most companies (small to medium and a large number of industries), that isn't a fulltime job. Most companies have IT as support or are a very small segment of the total business.