My general instincts are with Yglesias when it comes to leaving choices up to voters:
In general, I’m against restricting voters’ choices in terms of who might be president. I think parties should be allowed to nominate an immigrant if that’s what they want to do. Or a young person. Or someone who’s already served as president for two terms. Why not?And I'm against term limits pretty much anywhere else: Congress, state legislatures, governors, mayors. For presidents, though, I'm much more ambivalent.
It's a close call, in my accounting. I'm not really concerned about the wasted talents of Bill Clinton (who, after all, is the only term-limited president since the amendment was adopted who would have been a plausible candidate for a third term -- and before FDR, it's not as if there's a long line of presidents who would have won a third term if not for the norm against it). I do, however, believe that it's a bad thing to have second-term presidents who don't have to worry about re-election. Returning to the voters is a big part of representation, and that's missing for second term presidents.
On the other hand, as much as I like to emphasize how limited the power of the presidency is, and as much as I don't really think that incumbency advantage is a big deal when it comes to presidential general elections...well, I do understand the point of a two-term limit as a check against presidential power. After all, the president is certainly the single most influential person over policy, including some very potentially nasty policies. Starting, of course, with war, but also including federal prosecutions, detention and imprisonment, spying, and others. One big check on presidents doing awful things with that influence is the question of re-election. But suppose presidents could build an incumbency advantage of, say, five to ten points. That is, an advantage just for being president -- perhaps because of the way that the news media treats the presidency, or through the ability to direct resources to key constituencies, or perhaps through large-scale constituent service. It doesn't bother me much when that happens at the House or Senate level, but it does worry me if presidential elections became unmoored from the consequences of presidential action. The downside consequences for democratic control just seem a lot more serious.
I'm really not sure where I come out on it. Really, I'd like to return to a world in which a third term was Constitutionally permitted but strongly discouraged by norms within the political culture, but that, alas, is not available.
My favorite thing about the 22nd Amendment, by the way, is that it was passed by Republicans to punish FDR. That'll show him!