How do you define "source"?
"Source" IMO = the earliest-generation *complete* copy of the film representing the director-approved final copy that can be used for duplication.
Normally with modern films that means the DI/digital intermediate. With older films that may mean the photochemical interpositive or approved release-print.
In the case of Fellowship, the earliest-generation "copy" of the completed film was the photochemical print-master which was what was used as the source for the theatrical release and was the final "master" incorporating the intentions of all artistic parties: director/cinematographer etc.
Evidently the small FOTR recap at the beginning of TTT is much less soft than FOTR itself. So it seems that not all hope is lost for a potential future remaster.
This has been covered again and again.
The original camera negatives are sharp, and these were re-scanned for the later two films with all post-production work being done with better software and utilizing a final DI master. That's why those same scenes look sharp in those films.
For Fellowship the post-production software was in its infancy, so all effects processing really took a toll on image clarity which is one reason why it looks so "flat" most of the time. But the "source" for the movie is the *master*... not the camera negatives, and the final master for the finished LOTR film already had the softness locked-in to these scenes because of the post-production.
In order to fix that, you'd have to literally re-create the film from the ground up... or at least roll back a stage or two (or more) prior to the existing source master.
Think of it like an music album... the "source" isn't the live session takes/recording... it's the *master* made by the audio engineer where everything is mixed and modified according to his artistic taste. Even if the source session recordings would sound "better" to your or me personally, it doesn't change the fact that the *art* being presented to the listener is the final mastered mix. If they wanted reverb, or they wanted to play with effects, or they wanted to lower the volume of one singer in comparison to the guitar, that's their choice. That's the final art.
Movies are the same way. the camera negative is not the "source" to use as the reference, it's the final master that incorporates the final vision of the director and cinematographer.