Hhmm. . .
The initial storline, Superman: The Man of Steel, holds up very well. The reimagined origin, familiar, yet different, was a pitch-perfect introduction for new readers to the Man of Steel. For long time readers? Not so much. It demolished everything that had been built over the near 50 years prior, and much had been built. Still, DC didn't need to hold on to those readers with the new batch that came in. It's why they were able to stay the course with the 1986 rebot, and little else.
Superman and Action Comics were solid books, but I found them staid and unentertaining, and stopped reading. Superman, as presented by Byrne was missing a dynamism that I knew he possessed from watching the cartoons and seeing the films, and I went look for and found it the Bronze Age back-issues. I just didn't have an interest in reading about an Earth-bound, chained to metropolis, Superman.
However, there were still some great stories, and one has to remember that in less than 2 years, Byrne delivered over 70 stories for Superman. That is a lot of development. The World of Krypton miniseries is a great example. It was an excellent rendering of the planet Krypton: the time before and leading up to its fate. The shame in its execution is that Superman was made so distant from his Kryptonian heritage and so intellectually disinterested overall, it didn't play much further than the mini-series until much, much later.
The crossover with Superboy and the Legion of Superheroes is still one of my personal favourites, even though DC slammed the door shut and hard on any possibility of Superman being Superboy until Geoff Johns' Secret Origin. "The Greatest Hero of the All" always makes me tear-up on the inside.
Superman Annual 1 with Titano was great fun, and Action Comics 600 gave those who wanted a Superman and Wonder Woman pairing just that. . .with stout reasons why it would never work.
For the current Superman, Byrne is the man who brought them to the party, and they will always look on the work fondly, if not critically, as we tend to dance with the one who brought us. The Byrne issues aren't horrible, but they aren't great either. They are strangely adequate and not much more.
The one exception is the artwork. Byrne's linework is still better than 90% of the artists currently working. Those pages hold up and always will.