We always hear that the CEO's primary tasks are to 1) make sure there's enough money in the bank to survive, 2) guide the vision of the company, and 3) hire and retain top-notch talent.
But we rarely hear about part 2 of the CEO's job, which I think should be the next most important thing for a CEO to do: Make existing staff more productive, and to find a systematic way to get new talent up-to-speed faster.
It's certainly something that any startup CEO should be thinking about. However, it's arguable whether it should be one of the core responsibilities of a CEO.
Core responsibilities are those which, if not fulfilled, will kill the startup sooner or later. Making bad senior hires can kill your startup. Running out of cash will kill your startup. And having no direction (and no active communication of that direction) will also risk your startup's existence.
On the other hand, having an inefficient onboarding process probably won't kill you, unless it causes you to lose great people, run out of cash, or lose track of the company vision. This is why the other three are core responsibilities.
As for "things that the CEO should do as well", that entirely depends on the size of the company. At size 1, the CEO does everything. Even at size 2, 3, or 5, everyone is doing a bit of everything, and the CEO does need to care personally about every aspect of the startup. But, as a company grows, it becomes more important to let people do their job and focus on the three core CEO responsibilities.
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