Top engineer with academic or industrial or self-employed experience, whatever it be, some people are but talented to find solutions and can mold in at levels of problem solving. It is analyzing the problem and finding the solutions which can be implemented by junior level engineers or even themselves, if they are in startups or small firms, wherein one person has to take on many roles. So, the answer to this question depends on the ecosystem where one person is employed in. If the ecosystem is huge, then there shall be one person for each task, that is what happens in large enterprises. There is one person, who takes care of what all project runs in his unit, this person is called Engineering Manger of that division, and he reports typically to the CEO of the company to direct right funds of the company to his division projects. The Engineering Manager, looks at defining the problems and motivates people under him, such as Design Engineers, Consulting members and even project leads and makes them find the right solutions which are implemented by engineers.
However, in smaller firms, this ecosystem may be slightly smaller, in mid-sized companies, the hierarchy is more direct, the engineering manager keeps meetings with all members in a group. Typically, there are not many groups in small sized companies, hence often one Engineering Head is there, he is present in all kinds of meetings, with team and not just attends meetings with the team leads, consulting member, program managers and design engineers.
In start-ups things are even more different as the ecosystem is even smaller, here one person may have to take up the role of the software engineer after he has modelled the problem and then found the solution too. And as the startup grows, the junior level roles are filled up, and hence easing the implementation from the architects who had been doing multi-tasking of roles all this while.
Hence the answer to this question-of where does a top engineer spends most time is relative to the ecosystem of the place where he delivers work. But mostly, the work of top engineer is finding solutions and choosing the best one to be implemented by the junior engineers. The problem definition is depended on the goal of the organization or funding agency and top-level engineers can always participate to guide in the organization whether the problem has been rightly framed and what are the possible hurdles that can be obtained in the way to implement it, what can be the possible costs to implement it, the incurred costs to test it and how to send it live to the users of the product that the problem shall develop into. The problem definition is a relatively rare activity in non-large-scale organizations, the reason is – the smaller the organization, the lesser the projects, so once the projects are decided, the problem definitions are freezed. And now its time to find solution to bigger problems, then smaller problems, however the top engineers should have capability to double up to define new problems as and when it is deemed and required. This much is expected from a top engineer and he should be ready to find solutions and chalk out which one is best to be implemented.