Why do leaders fall to the perils of the Bathsheba syndrome?

Because they are human beings. Part of the lesson from the story of Bathsheba is that as great a leader, warrior and King as David was and as close to God as he was, he was still able and willing to sin. The lesson is that much lesser men will fail us to and to not put our faith in them because they can (and likely will) disappoint us.

"Absolute power corrupts absolutely" IMHO folks with that amount of power, obviously not all of them, possibly reach a point where they feel they can control everything. They become victims of their own supremacy because they feel nothing can happen to them or they are too far above the ground to be taken down by a mere indiscretion. Now throw the instantaneousness of social media into the fray where the verdict is reached in the court of public opinion before the facts come out.

If the likes of MacArthur, Patton and Eisenhower were around today in the media frenzy world we live in, would their faux pas (not all of them being sexual in nature) have allowed their careers to survive? Mac was ultimately fired yes, but would it have been done sooner? Ike had an affair and became president, would he have? Would Patton have been relieved permanently over the slap?

One would imagine powerful people would practice more prudence because of their lofty positions, but they are enticed with power, alcohol, money, and sex. I think they commit these deeds not only because they are human, but power has them drunk, and they perform a flawed risk management. There is a slim chance of getting caught despite catastrophic results. They probably wish away the risk.