An alternate 22nd Amendment

What if instead of forbidding Presidents from serving more than two terms, it simply forbid them from serving more than two consecutive terms? I don't think that it is totally ASB, since former Presidents have in the past tried to seek a non-consecutive third term (Teddy Roosevelt and US Grant). The tradition as I understood it was only that Presidents not seek a whole bunch of terms consecutively because that seems to close to monarchy for Americans to be comfortable with. Until FDR, no one had even tried for a third consecutive term. So what would be the affects if they had simply outlawed Presidents from serving three consecutively and allowed them to attempt comebacks after leaving office? Here's what I came up with off the top of my head:

Only four Presidents have so far been prevented by the 22nd Amendment from running again: Eisenhower, Reagan, Clinton, and Bush. Eisenhower would have been too old to have attempted a comback in 1964, plus I think he was experiencing medical problems already. Although if he attempted it, I'm sure he would do quite well. Reagan, pretty much the same. Too old in 1992, and even though he wasn't diagnosed with Alzheimer's until 1994, he was probably already feeling the early affects. Nonetheless, if he did run, he'd probably do okay. With Clinton it could get interesting though. He always loved the spotlight (like most politicians do). The Democratic field was relatively weak in 2004, and while President Bush's approval ratings had not yet gone into the tailspin that they did in the second half of his second term, he wasn't terribly popular either. Clinton is also fairly young (the same age as Bush), if he wanted to, he could conceivably get the nomination and beat Bush. Of course, he hadn't had to run for anything since the Lewinsky scandal, so I would be curious to see how that would factor in. Also, if everything goes as OTL up until now, Bush would be in an interesting spot right now. People in the south keep putting up billboards that say "Miss me yet?" with a picture of GWB on them, and a lot of conservatives, while not thrilled with Bush when he was President, are now longing for a few years ago. A "Draft Bush" movement, while probably ultimately unsuccessful would be an interesting twist in a 2012 race.
 
I doubt Nixon. In 1980 he would be 67, and though he was quite healthy to the end of his life there'd be questions about his age. Particularly if the Dems nominate a younger candidate.
 
He'd have run in 1976, though, and as popular as he was, he could probably get some public support, but breaking tradition would draw some blood.
 
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