Well, the President wasn't really supposed to speak for the people, but for the Union.
I think one of the greatest, most toxic misunderstandings of American politics has been the elevation of the Presidency to some sort of mythical overlordship of the nation. He's not. He's really just Congress's agent for carrying out policy in certain domains relating to the Union as a whole:- Treaty negotiation
- Trade negotiation
- National security
- Waging of war
- Execution of federal law in federal jurisdictions
Beyond that, his only real superpower is the "bully pulpit"; i.e., his celebrity status and resultant power as an influencer of public opinion.
Which was much more significant in Teddy Roosevelt's time, since there weren't that many celebrities who could command the resources and cachet of the US presidency to get their message out.
But we live in the Information Age. There are dozens, if not hundreds, of celebrities with a comparable pulpit, and comparable influence over public opinion. Everyone from Oprah Winfrey to Julian Assange has a pulpit just as bully as the President's.