How is this different from private business?
I've worked for lots of companies where the CEO and other executives use their personal email accounts almost exclusively. Top execs usually aren't reading or writing their own email anyway, at least in the places I've worked. The CEO's assistant does that -- they're the ones using the CEO's email account to read and respond, and usually are told either directly or from that personal email account what to do.
Not using your official email account is also a good way to get around e-discovery of your personal life. One place I worked for was so spooked by e-discovery that they set email retention for everyone to 30 days...if anyone came looking for evidence, they had better do it quickly. With CEOs being public figures, likely sued many times a year, they might not want their official email accounts to contain details of their...personal lives...that wind up in front of a jury.
Government is a little different, because top Cabinet posts probably have the same rules requiring archive and retention of official communication. It sets up an interesting approval circular reference when your manager has to approve IT requests...seriously, does the President do that? :-) Who approves his IT requests?