How sound is the Hadith "Fast and you will be healthy," and how credible is the website "Dorar" which stated that the Hadith is not authentic, and why do preachers and imams cite it if it is not authentic?

The aforementioned Hadith is weak or flimsy (wahin). It was deemed weak by Al-Hafiz Al-Iraqi and Al-Munawi, and was included by Al-Saghani, Al-Fitni, and Al-Shawkani among the fabricated (mawdu'at) narrations. Al-Uqayli also narrated it in "Al-Du'afa'" (The Weak Narrators). Furthermore, Al-Albani and Ali Hashish declared it weak. Despite this, Al-Mundhiri attributed it to Al-Tabarani and stated that its narrators are trustworthy, and Al-Haythami concurred. It is important to note that the mention of a Hadith by preachers and imams does not necessarily imply its authenticity, as many preachers do not meticulously verify the authenticity of chains of narration, and some are lenient with Hadith concerning virtues. It is permissible to act upon a weak Hadith in matters of virtuous deeds under specific conditions.