Did Dovid Ha-melech sin?
Well, he said "חטאתי" after Nosson told him the משל כבשת הרש. He was also VERY severely punished [see Rashi Shmuel Aleph 12-6]. So something was off. According to one opinion in Chazal he didn't sin the way the psukim make it sound. According to other opinions in Chazal he did sin very badly. [See Sanhedrin 107a and Ksubos 9a]. The Abarbanel very strongly asserts that he sinned as the simple reading of the psukim describe.
That being said he remained king. Why? Because he didn't try to justify himself. He just admitted his wrongdoing and did teshuva. Shaul, on the other hand, tried to justify himself of his indiscretions and as a result lost the מלוכה. [See Yoma 22b]
Big people sin and big people make mistakes. Nobody was bigger than Moshe and if you were in Israel this Shabbos you heard about his mistake in shul [this coming Shabbos in חו"ל]. No cover ups. No paying loads of money to PR people to help clear your name. Just the bare facts as painful as they may be. We don't rewrite history to make ourselves feel better. [I for one would be much happier had our great leaders never sinned].
The halachic definition of לא תחמוד [number 10 on the BIG TEN list] is that one not only covets one's friend wife but takes measures to get her [see גזילה ואבידה א-ט]. That is a REALLY bad aveirah. If a Rosh Yeshiva or Rav does this aveirah [based on Ruach Hakodesh of course...], then it is very hard to excuse. UNLESS, he publicly admits his sin, acknowledges the חילול השם he created [for which there is no teshuva in this world and only death can atone - Rambam Teshuva 1-4], and makes amends.
But when he makes a video presenting himself as a poor, innocent victim of evil people trying to destroy him [after a Beis Din found him guilty and he admitted it to them] while admitting no wrongdoing, it is a scandal. It is such a scandal that for the sake of the Name of Hashem it must be called out.
As Chazal say: במקום חילול השם אין חולקין כבוד לרב.
One of the basic principles of the Torah is קדושים תהיו. Someone who is a teacher and guide to thousands has an added responsibility.
The above may or may not be referring to real events.....