A CEO taking a day out of their schedule to shadow "an employee" makes for viewer counts on TV, but it can't give them any insight they can personally action on, and it can't give them the several different perspectives they need to properly action on without doing _lots_ of shadowing of various employees in various positions: so this is why there's a management and feedback structure. If there are things the CEO should learn, they should be able to rely on their executive team to bring that up.
If you're not a CEO, you can learn a lot from shadowing your CEO, without impacting the viability of your company. But if you're the CEO, all you can learn is "how something seems to work at a level you are not supposed to have control over." Unless your company has an incredibly flat org, without a board that the CEO is accountable to, and where the CEO is actually makes direct hiring, product, etc decisions, realistically having the CEO shadow employees means that they're not being the CEO. The larger the org, the more people they'd need to shadow to get meaningful insight, and the more the board will go "we need you to be the CEO of this multi-hundred-million dollar company, so if you want insights: you should already have a team in place for that."
It has been the case with most companies I have consulted with, as a management consultant that this is rarely the case.
The CEO and his favorite executives and VPs salesmen live inside their own little bubble with little real knowledge of the problems and challenges that the company is facing internally.
This is why I made good money.
It is easy to see this when management sends out decrees on how to improve
performance or throughput .. etc that does not contain any mention of, nor solutions to the problems that is keeping the performance / throughput /(or whatever fitting metrics are for the company)
I might have a bias in that I have mostly worked with companies that hired us to help them. It is possible that the pool I have been exposed to is different than the majority but I do not think so.
I know of companies that are run in a more sane way but they are usually smaller, at least in number of employees and thus leadership is more aware of how things work.
CEO shadows an employee all day. Get a feel for how the peons live.
Understand why it is so hard to get things done in the company.