What is the ruling on a Hadith that did not meet the conditions for acceptance yet was received with acceptance by the Ummah, and what is the evidence for this from the sayings of scholars, with clarification through two examples?

The conditions for accepting a hadith are five: uprightness (adalah), precision (dabt), continuity of the chain of narration (ittisal al-sanad), and freedom from shudhudh (aberration) and illah (defect). A hadith that fulfills these conditions is considered acceptable (either sahih or hasan). A hadith may be judged as weak if one of these conditions is not met. However, some weak hadith are acted upon due to their widespread acceptance among scholars and their conformity with Islamic principles and rules, such as the hadith of Mu'adh regarding the method of judgment, and the hadith "Every loan that draws a benefit is usury."