Isn't it a fundamental startup-culture tenet that the CEO should be a "whatever needs to be done" actor within the company?
To borrow Spolsky's yacht analogy, the captain has 40 people running around below deck to ensure that the engine keeps running, the beds get made, and the meals show up on time. You don't want your customers (your developers and your designers) to have to think about anything besides where they want to go and what to have for lunch.
But it'd be a pretty inefficient boat captain who tried to rebuild the engine and cook a 5-star meal at the same time.
I think you're taking this too far.
your statement is true if they have nothing else to do.
However, if the company needs money and the choice is between your CEO fundraising or answering emails then I'd rather have my CEO fundraising.
There are only so many hours in a day and you can't expect your CEO to do all tasks that no one else wants to do.
Is this allowed in the US? In several European countries, legislation on interns is very strict: they need to have a specific project to work on, that is not part of the day-to-day functions of the host company. This is to stop companies depressing the wage market by using unprotected interns instead of salaried workers.
So you can take an intern to design and launch a new recruitment process, or other "project" work, but not just to offload recurring tasks.
It's the same with client relationships: we may be neck-deep in server fires at the moment, and know there's no way we'll be able to fix something today, let alone before the end of the week, but simply having someone reply "Yes, we'll handle this, and I'll take care of following up with the development team for you and get back with you shortly" goes a long, long way to putting a client at ease.