I can think of at least three real-world advantages we gain by learning how to cite articles manually:
It compels us to become familiar with the syntax and form of citations, which makes references to articles easier to read and parse ... something that we all still have to do manually.
If you only need to jot down or reference one article you can do so without troubling with software tools.
By knowing how citations should be formatted, you are in a position to recognize errors and correct them (i.e. you can tell when your bib software malfunctions).
Pocket calculators are cheap, reliable, and widely available. Why do we insist on learning addition? The answer is, there are a lot of times you might want to use addition and you don't always want to depend on a calculator for that.
Of course we should take advantage of software tools for compiling references and bibliographies, but this is not always viable or always necessarily better than doing it manually unless we can assume that bibliographic software:
- correctly cites articles in the desired format.
- composes references with little effort (automatically).
- is freely available to everyone (i.e., at no cost).
In my experience, all three of these assumptions break down much more often than one would like. For instance, it might be difficult and very frustrating to make a small change to a standard format in your bib software, which is required by the journal or funding agency you need to work with.