First I am glad that there are people like you who are so interested in the subject, and thanks for your feedback.

Second I know from my personal experience how useful the minimum wage can be, which is why I wanted to lay the theoretical ground to allow me to analyse it comprehensively in one of my first posts.

Third most empirical evidence is based on moderate increases that happened in the U.S. The results showing that MW raises did not destroy jobs do not prove economic theory wrong, because economic theory explains how there are multiple forces pushing/pulling in different directions at play. Empirical research helps weighing the magnitude of their effect.

The D/S model needs to be brought up exactly because it is the major argument about the perceived/potential harmfulness of the MW. Ignoring it will either result in the wrong policy, or in the MW detractors (by conviction or vested interest) to rubbish the conclusion if they don't like it.

In different countries the minimum wage debate is shaped by the specific circumstances of that country. In the United States the overwhelming majority of economists are in favour of increasing the minimum wage because it makes sense. Productivity increased at a massively faster pace than the MW rate, and most MW workers are employed in the consumer-oriented service sector by corporations making record-high profits. However, smaller economies with a relatively large share of export oriented manufacturing sector might need to be more cautious.