> Isn't it a fundamental startup-culture tenet that the CEO should be a "whatever needs to be done" actor within the company?

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.