On the contrary, I think it's very republican (small "r").
First, anybody gets stale in that job, usually about a year into their second term (if you want to promote a single term of 5 years for President, I'll sign onto that). Basically, being President is such an intense intellectual exercise, it's mentally exhausting. You use up a lot of your intellectual capital during your tenure, with no break to recharge it. You need to relax, to read and think, to write, and to exchange ideas and concepts with friends and allies. A President in office cannot do this -- he's surrounded by yes-men or people who want somethig from him. This effect increases in severity with time.
Second, we no longer elect a man as President, we elect a team of people. The same intellectual bankruptcy affects them as well, to a lesser degree, but more than that, ways of doing everyday business become ingrained, usually to the detriment of effective operations. The electorate is largely oblivious to this effect -- they just see things getting worse with time and assume it's the President's fault. In part, it may be, but mostly I think, it's the fatigue of the job on the myriad of people who make up the entire administration.
Third, nobody is indespensable. Presidential term limits re-enforces the idea that this job is not a career (or should not be, anyway) and that Presidents should take office, lead and accomplish one big thing or a few lesser things, and step aside. To me, that is the very essence of a republican tradition -- service for the common good and then stepping down. Eight years is enough for (and in some cases, of) anybody.
You hit the nail on the head. The fact that it should be such an intellectually exhausting exercise, is reason enough to squelch a third term for anyone, Democrat or Republican. Most CEOs can't sustain repeated success within a company; we shouldn't expect that any one administration can sustain long-term success with the rigors of a presidency.
There are relatively few CEOs in the corporate world who continue to be effective after as long as eight years on the job. Let's face it. If you can't implement your agenda in eight years, what makes any sane person think that another four or eight years will do the trick?