On all of your reputation issues, people's judgment of the quality of doctors is almost entirely a function of their bedside manner, and not their competence. If you just go off of reputation, and reputation is based on the decisions of lay people, the result is that competent doctors get bad reputations and confident charlatans get excellent ones. And health suffers accordingly.
A similar well-documented example are public health measures. My surface impressions of a restaurant are a bad guide for whether the cook washes his hands properly after going to the bathroom. And dysentery comes through infrequently enough that you won't discover this from your neighbor getting sick until far too late. But well-constructed licensing requirements solve this problem and save lives. (In fact there is good data suggesting that public health measures like these save more lives year in and year out than all of the advances of modern medicine.)
We also know what happens with public engineering. Before licensing requirements were created, the USA averaged one random bridge collapse per week. That's pretty bad. It gets rapidly worse if you get an earthquake. I live in an earthquake prone zone, and I am a very firm supporter of having the buildings I am around and in be built to standards created by qualified engineers, then inspected by properly licensed inspectors. California is among the few areas where significant earthquakes don't tend to come with massive loss of life, and I appreciate that fact.
I could multiply examples, but the flip side of the argument does not hold. The article is correct that there are a lot of stupid licensing requirements out there. The article is correct that the motivation is often restraint of trade. The article is correct that we're often all better off without those rules. But the article is dead wrong in concluding that we're better off removing all licensing requirements.