Taking control audition style may work if you're a consultant trying to convince a client to hire them, but you had better know their business backwards and forwards, and unless you're in a very well defined space (like their football coaching example), you probably don't.

However, if you attempt this at a standard FAANG job interview, it is emphatically not going to work. There's a set of questions you need to work through and attempting to go off script is not going to be appreciated. Pretty much the only exception is leadership roles, particularly when the brief is "we need somebody who can turn around/10x our floundering business".