Running day to day business is actually the COO's (Gwynne Shotwell's) role:
Chief operating officer - Wikipedia
"The chief operating officer (COO), also called the chief operations officer, is one of the highest-ranking executive positions in an organization, comprising part of the "C-Suite".
The COO is responsible for the daily operation of the company,[1] and routinely reports to the highest-ranking executive, usually the chief executive officer (CEO).[2]"
A CEO's primary role on the other hand is to lead and be visionary, among other things:
Chief executive officer - Wikipedia
"Typically, responsibilities include being a decision maker on strategy and other key policy issues, leader, manager, and executor. The communicator role can involve speaking to the press and the rest of the outside world, as well as to the organization's management and employees; the decision-making role involves high-level decisions about policy and strategy."
The whole purpose of a COO is to offload the day-to-day management responsibilities from a CEO.
To suggest that Elon isn't the CEO of SpaceX is false, disingenuous, intellectually dishonest and bordering on a lie.