The Founding Fathers never expected people to make a career out of elective office.
I think that's partly true. But the way in which the U.S. government borrows from its predecessor sort of hints at it. The Senate is clearly analogous to the House of Lords, an unelected cadre of elites that nevertheless believed themselves to be born and bred for leadership and held their title and office for life. The six-year term of Senators and their special role in approving the actions of the
sovereign—er, um, executive—spell the intent for this body to be the sober, deliberative, born-and-bred branch of government. Ideally we are supposed to elect to the Senate people who might make good lords and ladies in the House of Peers.
Conversely, there was a lot of concern about the damage the unwashed rabble in the House of Representatives could do. Hence the two-year term. Certainly the Representatives were meant to be a more pure republican expression, chosen from rank-and-file Americans. (Here in Utah, our state constitution mandates a lay legislature and limits their sitting to 45 days per year.) The 6- and 2-year terms were the original term limits. Politicians and diapers should be changed frequently, and for the same reason.
Instead of term limits, perhaps a retirement age would work better. Elect that person as many times as you want, until they're too old to be effective. The pressing problem with Feinstein and others is not that they've been repeatedly elected, but that they're no longer capable of performing the duties of the office.
Now you can argue that the practical requirements of modern national government require elected officials with professional commitment. I'd probably agree. You can point out that the U.S. Senate has ceased in its role as a body above the fray. But if the question is whether the Founding Fathers designed the government to require more than lay education, experience, and commitment, I can answer confidently: maybe?