I think you're almost onto something here, but I think the problem is not compensation or benefits. It's that the President has too much power.

And if I may quote Spider Man, with great power comes great responsibility. The constitution wasn't written under the assumption that the federal government would ever be as large as it is. It isn't set up for it. So the President is in charge of too much and is consequently spread too thin for any one person to handle.

Here's the problem: The number of officers of the federal executive branch who are elected: One. Just One.

Compare that to anything else. In any given state you'll elect the governor, but you'll also elect the mayor of each city and possibly the sheriff, etc. I want to vote for the head of the FBI. I want to vote for the head of the IRS. Stop putting it all at the feet of the President.