At my current job, a 10 person startup, the CEOs job is to generate leads, talk to potential customers, and close deals. He, along with the other founders, shape what we work on based on what he thinks it’s possible for us to sell. He’s essential to the business.
At my previous 50k person company I never talked to anyone within 4 levels of the CEO and thought he was useless. But realistically he probably had a somewhat similar job - talking with the board, talking with politicians, and talking with the CEOs of other companies in the effort to lower costs (tax breaks, partnerships, etc) and generate sales (contracts, partnerships, etc).
Similar, yes, but instead of finding out who else can give you money at 10 people, they have to find out who else they have to give money at 10k people. Leads and lobbies.
That's a sales person.
A sales person is usually trained on an existing product or portfolio of products (including services). A good sales person might use their exposure to customers for gathering valuable information to feed back into their upstream org. But this is completely different from someone who has to actively shape a both sides of the company, value delivery and value creation, because both need to mature in a choreographed manner.
So you are right in one direction: a good CEO often has to be a good sales person. The reverse does not apply, though.
There’s a reason that many CEOs have a stint in sales.
In a stable, profitable company, you need sales to keep the business solvent. And understanding the sales pipeline is pretty essential to effectively running a business.
Where this breaks down is when the C-suite forgets that you also need a product, and you need to keep innovating on that product in ways that bring value to current and future customers. This seems to happen pretty frequently.