From a chemistry point of view, if dental amalgam is a compound of mercury and other metals, and compounds are elements which are chemically bonded, how would it be possible for mercury to "leak out" of fillings if no further chemical reaction would be taking place?
You are labouring under a misconception about the exact meaning of the word "compound".
Normally when we describe, for example, ethanol as a compound of hydrogen, oxygen and carbon we don't mean that the component elements can "leak" out of it without a chemical reaction taking place.
Mercury amalgams are casually described as "compounds" but not in the same sense as ethanol. Amalgams are mercury alloys with varying compositions but they don't, usually, consist of strongly bonded molecules like ethanol with strong chemical bonds preventing the molecules falling apart. They are closer to mixtures than what we normally think of as compounds. Having said that they are not just mixtures as they can have very different bulk properties than their components.
The issue with mercury amalgams used in fillings is that they are mostly, but not completely, inert. Some mercury will evaporate from them and some chemical and physical processes can increase the amount that evaporates. But good estimates suggest only micrograms/day which is less than typical workplace exposure limits so probably not a source of legitimate worry. The dentists who still use mercury fillings are likely exposed to more than this and this is a better reason to limit their use. The subject is controversial with a lot of disagreement among studies. This is well summarised by the Wikipedia article on the controversy.
Mercury vapors are released. It has long since been proven that people with mercury in their teeth constantly have small portions of mercury vapor evaporating from the tooth. There's many videos of this taking place. It's not a mystery.
Why do some dentists claim mercury “leaks out” of amalgam fillings?
Because it does.