I think you're missing the point here.
How do you get to a profile to connect to someone without using the search?
You can't - unless you know their username and can add it to the URL. If you run a site where the users are less technically capable than a houseplant, then this isn't feasible.
Asking them to input their friends username into a box is the easiest way.
As I mentioned, a specific solution to a specific problem.