The label is used as a parallel to the "executive functions" of the brain. This layer control the behavior of the overall organism: it selects goals, and monitors behaviors required to attain those goals. This describes the intended role of (upper) management inside a complex interdependent organism that is a business with 1,000+ independent brains working on it.

An executive isn't the same as a manager, but decide what the business shall do (and then do it/execute it; usually this means hiring people though). They are accountable only to the owners of the business and legislation (and maybe only the latter if they are also the owner). An executive doesn't have to be a manager, for example the programming CTO who has no employees and writes all the code themselves. A manager can also be an executive, for example the CEO who manages their direct reports. A manager might not be an executive, for example, if their boss is another manager

My interpretation is that these are the group of people who "execute" the will of the board. The board are the ones who set the direction of the company, but then the Executive Officers of the company are the ones who execute that.

Ah, you are missing the key thought that they do the execution by getting all the cogs in the machine to move.

There is probably some explanation, but I don’t think the etymology is any more transparent to native speakers than it is to you. It’s just one of those words whose meaning has drifted over time.