We are talking about different things. In all the cases you described, the president either stepped down on his own or was forcibly removed by outside powers.
The important point is “stays sharp”. If your president suffers from mental deterioration, he or she would have to be removed. In such a case you might have a president who actively resists removal and who’ll have active enablers, who’ll be helping him to stay president.
I’d call such a situation highly undesirable. Indeed, we seem to be close to witnessing it right now, the next years may get even more interesting.
Pope Benedict had the wisdom to step down. But it may well have played out differently.
