There is Artin's approach, and then there is a more abstract approach by Grothendieck relating to the fundamental group of algebraic topology.
In the movie Beautiful mind there is a scene, where a student tells John Nash that he can proof that 'Galois extensions are the same as covering spaces'. This follows from the Grothendieck's approach. However the analogy between Galois extessions and the fundamental group was known even before Grothendieck.
Then there are even more general approaches in the category theory setting.
Today these generalisations are taught indeed without much regard to the computational spirit of 19th century mathematics. They have their merit as you say, but I agree understanding the computational aspects are instructive in fully appreciating the generalisations and analogies.
Very true. When I was doing PhD in pure maths I had random people explain to me that a PhD would not necessarily give me an advantage in the job market. Like I would come to do a PhD without figuring that out.
Not specifically in pure maths, but agree I also found that a math PhD is good to have for jobs in AI, finance etc. But an equivalent industry experience might be worth the same at least.
For me, data science, quantitative finance, software engineering jobs. A large part of software development is about identifying and building abstractions to solve problems, which is also what mathematicians do. A common sentiment I've seen expressed by hiring managers is that someone with a strong math background can become very competent in a wide variety of technical fields.
Samuel Eilenberg, one of the founders of category theory, used to tell to his students "You should have you own category". Meaning, that when you work on category theory you should have in mind applications to something. One of his students used to say "my category is Cat". Cat means the category of categories.
"Algebra" by itself doesn't appear on this map. Clicking through and reading some of the descriptions, my impression is that this was not created by mathematicians.
It is not true that traits that are not advantageous are unlikely to be maintained over many generations. For example, many genetic diseases have survived for many generations. This may happen because, for example, they correlate with some other more positive traits. Or, because of other indirect, external factors.