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.
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.