There is never one person who 'designs an aircraft'. The first concept sketch may be done by one person, but turning that to an aircraft takes hundreds or thousands of people. The "chief engineer" or "chief designer" is essentially a technical manager, responsible for the success of the program as a whole, making decisions.
Begin definitely was heavily involved in the F-5 configuration development along with Gluyas and others. Gasich was their boss, in charge of the Preliminary Design department, and a strong influence on the F-5. Schmued was Gasich's boss, and by this time squarely a management guy, though he did still have technical influence.
An Air Force guy commented on the N-156 rollout "you tell its a Schmued design", which amused some of the Northrop F-5 team including Gasich at the time knowing how he'd had to fight to get the engines in the fuselage against Schmued's advice.