Hazlitt uses the whole book to drive home the most important lesson about economics, and that is: One must consider not just what is seen, but also unseen.
Let's apply this to the case for minimum wage for hotel workers. Sure, the minimum wage helps hotel workers. Sure, hotels will still pay for cleaning services. That is all seen. That is the naive analysis. What is unseen? What is the unseen result of a higher-than-market hotel-cleaning wage?
Workers who would best be employed in other tasks will now be employed cleaning hotels. Other goods and services that are more valued won't be produced because the people who would produce them are busy cleaning hotels.
In general, the low-productivity employment sector will be made less efficient, because of lack of price information. For example, suppose the minimum wage is $7/hr, and the market wage for hotel cleaning is $5/hr while the market wage for cooking is $6/hr. A worker who could do either should, in a free market, choose cooking, because it is more valued (hence the higher wage). But a minimum wage hides this information. So if the worker chooses cleaning, the economy will be $1/hour poorer than it would otherwise have been.
Also, the sector of the economy that employs large number of low-wage workers will become comparatively less efficient because of the high-than-market wage costs. This will encourage capital to flow out of this sector and into other sectors. The unseen effect of the minimum wage is a missing low-wage industry.
This was made very obvious when the US Congress raised the minimum wage on the small pacific island of American Samoa. The result was that major employer on the island left the island and instead built a more mechanized plant in the state of Georgia. A lot of low-tech, low-wage jobs in American Samoa disappeared and were replaced by a lot fewer, higher-wage jobs in Georgia. This was the direct and visible effect, but over time, the unseen effect is the same: industries and jobs that won't exist because the minimum wage makes them illegal.