Most of this is just Asami acting like a normal friend, things that any other character on the show would do. You can't complain that people never saw subtest on it, Sokka and Toph had way more than it, and they had less than 5 minuts of "shipping moments" on the entire show. Do you deny it?
Didn't helped that Asami had less personalitty than Appa, and her interactions with Korra were bland, like "Hello Korra, you look great today." "Oh Asami, thanks for the complement, you sure are a good friend".