Per Dictionary.com, definition one of the verb “qualify” is: “To provide with proper or necessary skills, knowledge, credentials, etc.; make competent.”
“Competent” is defined as “having suitable or sufficient skill, knowledge, experience, etc. for some purpose ... .” It is no false analogy to state that someone who has demonstrated a high level of proficiency in skills including but not limited to leadership, management, delegation, decision-making, negotiation, and public relations in the private sector is a qualified candidate for a public-sector job that demands a high level of proficiency in leadership, management, delegation, decision-making, negotiation and public relations, among other things.
This is not to say that possessing these qualifications invariably makes one the best candidate for the job, or even someone who should be seriously considered for it, or that a given fast-food burger-flipper might not turn out to be a better chief executive than a given company CEO. And one can argue that the presidency presents challenges for which no other job is adequate preparation.
But “qualified” does not mean “best” or “ideal”; it means that the CEO can produce evidence of possessing a skill set that promises to be useful in the presidency. The two jobs are indeed analogous in ways that burger-flipping and the presidency are not.