O(n) algorithms for finding strongly connected components are referenced here
I think your intuition is correct. However, the wording "we can add at most 2k edges to make the graph a strongly connected one" is confusing. In many such graphs, you can add more
than 2k edges without
making the graph strongly connected. For example, if an SCC only contains a one-way cycle, all the edges in the opposite direction could be added without connecting any SCCs at all. The statement should therefore be that "up to 2k edges may have to be added to make the graph strongly connected". But those edges must be chosen the way that you describe.