This thread has taken a turn that i wasn't expecting, lol.
I think it's likely that when Carrie said what she admired most in Linda was doing whatever she wanted, she meant not being bound by supposed genre limits or preconceptions, and not being bound by what radio was currently playing.
Cover songs can help to illustrate that freedom. The two that Carrie sang in the tribute performance had both been revivals by Linda. "Blue Bayou" was co-written by Roy Orbison at the beginning of the 1960s. Thematically, it was pretty obviously inspired by Jimmie Rodgers' "Miss the Mississippi", from the beginning of the 1930. Both also drew musically from Black-White fusion styles. "When Will I Be Loved" was written by Phil Everly, also at the beginning of the '60s - even at the time, it could be considered a late example of Rockabilly (another fusion style, between Blues and Mountain Music), which had earlier entered the Mainstream, but was well past its heyday when the Everly Brothers recorded it. That would have been all the more the case, when Linda decided to revive it. This tradition of drawing on past songs, fusing styles, and extending ideas into new forms has been particularly important in Country Music (which, even at its most commercial, has never fully abandoned its roots, as a folk idiom). Linda's freedom was particularly evident in her choosing to interpret songs that conventional record company thinking might not have regarded as predictable, in terms of fashion, or current stereotypes. (The particular stamp she put on them often earnt them wide public and critical acclaim, notwithstanding conventional "play safe" opinion).
The general question raised about the importance of songwriting (and its relation to overall artistry) can never be settled definitively, because so much depends on the person being asked, the type of music being considered, and the relative weight attached to different elements in the overall creative process that brings us the music we listen to. Obviously, a classical enthusiast is likely to place much more emphasis on the composer than a Pop fan is likely to, for example. I would place Country Music somewhere between those extremes - fans who are more attracted to the stage performance and voice of the singer may be relatively closer to the Pop end of the spectrum, and largely indifferent to the name of the composer - while those whose tastes run closer to the Folk or Singer/Songwriter tradition may be more inclined to value artists as much for their writing. But, even in classical music, things are never cut and dried - the works are usually listed primarily by the composer, but the interpretation by the performers or orchestra directors, can vary greatly, and have to be given considerable weight. In Country Music, we often do value artists highly by the songs they have written - and songs by people like Hank Williams, Merle Haggard, John Prine, Iris Dement, Townes Van Zandt, Lucinda Williams, etc, are widely admired, revived and copied by other artists - but, again, it is not a case of 'either/or' - those artists are also admired for their singing styles (which is not the same thing as their technical vocal brilliance) and for their mood expression, and their interpretation of their own, and others' songs.
Instrumental music, by the way, adds another dimension to the discussion, because players may be valued more for the improvisation that they incorporate in, or add to, an underlying melody - far more than any actual notes that they may or may not have written. That is largely another topic - but it is particularly important in fields like Bluegrass and Country Rock.
In Carrie's case, I think there are certainly points of similarity between her approach and skills, and those of Linda. But I also think there may be areas in which Carrie might be considered more innovative. In my opinion, it would short-change Carrie if we only considered her as a great vocalist. She may indeed be the greatest of her generation, in that field - but, to my mind, it pigeon-holes or stereotypes her to consider that alone. Carrie herself is proud of her songwriting, and has been praised for it by people she's worked with. The production on "Cry Pretty" also shows that she both has an ear for, and is not to afraid to extend , musical innovation. An obvious example to me is the innovative way she has included progressive use of resophonic instruments, especially dobro, in some of the songs. To me Carrie is an all-round musical force in today's genre, and deserves to be considered for her creative skills as a whole.
