I won't tell you something new because you already know the answer yourself and all the greatest companies you see around been created by great tech talents / inventors. If you believe business is easy, you will just hire that talent, believe that that talent would be happy to invest his/her own time on you and something great will happen from that in the end, then it's called an idiotism.

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In the context of your question I will assume that by CEO you mean main co-founder in a tech company.

In any small-medium tech company CEO should code simply because he has main vision, usually has best skills and basically he can produce better and faster results without hiring expensive engineers.

If CEO of a tech startup is not a smart tech person, that startup will fail. Even if you will find a good CTO, which is very hard and expensive, you still won't move as fast as you could.

However, the main role of the CEO is culture, strategy, organization and business and bigger company becomes, less time CEO has on dealing with typical coding and if there is something you can delegate, you should delegate.

Apart from typical coding there is also research, data science, product design in general including software design and architecture. If you have great minds you can trust and everything is moving smoothly, you don't need to jump into the coding process yourself, you just "keep an eye" on it and only when really necessary you may solve some top-level complicated problems yourself.

If you are a great Leader who cares about a technology, product, quality, brand, research, company, customers and employees then you always will be part of that process.

All CEO's I've met where coding in the past and still doing it sometimes. My opinion is absolutely positive and even more - it is a requirement for me, I ignored all offers where CEO wasn't a tech mind simply because it won't move company anywhere in long-term and that person would never understand you, speak your language. Of course, when CEO is helping with dirty job (coding) it helps company as well.

Yes, many of them do, especially in tech companies - even Zuckerberg still does R&D coding to not fall behind.

Steve Jobs hasn't written a single line of code. It is not necessary to have a ceo who codes

As the company grows bigger, I suppose it's going to be harder and harder for the CEO to code, given the lack of time.

I think if you're running a tech start-up, it's always nice when the CEO codes from what I've seen, since they understand engineering issues and are usually on the same page with you. I've worked with non technical managers in the past and one of them wanted me to build our application's search in just 2 days.

That being said, trivial coding tasks should be given to engineers and the CEO could just take part in high level design/architecture discussions.

Our CEO does code, he comes from a network engineering background, and can put together a simple iOS app in Swift to show concepts as well. But it's not common to have such a guy as CEO.

Should they learn and code a little? Well, short answer - it's not necessary, but life would be much better for programmers if they did.