No, it absolutely makes sense here and was in fact seriously considered as one of the proposed versions of the 22nd Amendment!
I had a post on it in soc.history.what-if in 2002 (which I reproduce here with one link correction):
***
"No person shall be elected to the office of President for more than two
*successive* terms." (Emphasis added)
This is one of the alternatives that was considered when Congress was
debating what would ultimately become the 22nd Amendment. (See
https://groups.google.com/d/msg/soc.history.what-if/TSqO4QOQSJc/sI3YwOBRr50J
for my discussion of the legislative history.) This particular language
was voted down 50-34.
Suppose it had been adopted (instead of the language of Senator Taft which
became the Amendment as we know it). There are reasons why the Republican
majority of the 80th Congress might have found it attractive. After all,
like the amendment in OTL it was an implicit criticism of FDR--but unlike
OTL's amendement it would not be an implicit criticism of the Republicans
Grant and TR who sought third (non-successive) terms. (Of course maybe some
of the more conservative Republicans *wanted* a rebuke to Teddy as well as
Franklin.)
Apart from such considerations, this alternative 22nd Amendment, like the one
of OTL, would have answered the fear so many members of the 80th Congress
expressed--of a president using the advantages of incumbency to perpetuate
himself in power indefinitely. (Apparently, the thought was that only members
of Congress should be allowed to do that!) After all, when the third term
he seeks is not consecutive the ex-president does *not* have access to these
advantages. Of course one can imagine a situation where a president, having
served two terms, gets some "placeholder" elected to serve until he makes a
comeback. But there could be no guarantee that such a "placeholder," once
elected, would really be the stooge the ex-president wanted him to be, and
would not interfere with the ex-president's comeback.
So anyway, suppose this alternative 22nd Amendment is enacted. (Let's say
the language is tightened up to prevent any interpretation that a president
could resign in, say, the middle of his second term and then claim that he is
eligible to run a for a third term a couple of years later because it's not a
*successive* term.)
How would this change US politics? Some possibilities:
(1) Could Ike be persuaded to run for president again in 1964? His age and
health make this seem unlikely, but suppose the party leaders desperately
explain to him that only he could stop Goldwater, and that a Goldwater
nomination means an LBJ landslide and an overwhelmingly Democratic Congress?
("All those liberal big spenders will bankrupt the nation.")
(2) Suppose conservatives pressure Reagan to run in 1992? ("Bush is too
unpopular; he betrayed your legacy with his tax increase; he can't win a
second term--the party needs you," etc.) Could he beat Bush in the primaries
and then win the general election? Doubtful. To be sure, Reagan's
Alzheimer's was not officially diagnosed until 1994, but his deterioration
would be evident during any fairly long campaign.
(3) Of course there is always Richard Nixon, eligible in elections from 1980
onwards. (Remember, he was neither impeached nor convicted!) But I don't see
his comeback going quite *that* far. (Anyway, could Ford constitutionally
make Nixon's pardon contingent on a promise never to seek the presidency
again?)
(4) The most likely change is one that I suppose properly falls into
alt.history.future rather than this newsgroup--Bill Clinton would now be
preparing his 2004 presidential campaign. (Unless the very prospect of this
would lead to his being convicted by the Senate in 1999 in order to prevent
it--but that is very unlikely. Also, the Independent Counsel might take a
harder line on Clinton in 2001, believing him to be too dangerous *not* to
bring serious criminal charges against. But even if he gets Clinton
indicted, getting a jury to convict may be difficult, especially in the
District of Columbia.)
https://groups.google.com/d/msg/soc.history.what-if/zOIyaBkvlN0/CC78keinl3wJ