You'd have to be prepared to get rid of the entire concept of the "head of state". The USA has proven quite well that having the head of government and head of state combined has substantial problems. A head of government needs to be somewhat competent and deliver results, but a head of state needs to project "gravitas", "dignitas", and other virtues. In the case of the UK, the prime minister has been relieved of the need to project these qualities and be a "mascot", but a US president has not.

In theory, the people themselves are the source of legitimacy for modern governments. But the media and some of the population demands a figurehead that provides centralized moral leadership.