My test for composers is simply how I’d feel if all of their compositions would suddenly be lost. In other words, I look mainly at the best compositions, while taking into account also the number of good compositions. I don’t care if they also produced a lot of dross.
From that perspective, looking at the OP, I agree with Tier 1 (would be hard to dispute). Although for me Brahms is on the boundary between both tiers.
For Tier 2 I’d lose Chopin and Dvorak, who wrote music that I like but in my mind not really great music (with the exception of Dvorak’s Cello concerto). I’m not entirely sure about Haydn either: I find a lot of his work very good but not great. Possibly I’d add Mahler who I feel is more deserving of a spot here, and I’d consider Shostakovich and Vivaldi as well. Both have written a number of excellent works.
Tier 3 for me consists mostly of composers who have written only a few great works among many more mediocre or serviceable ones. Schumann and Bruckner, Elgar, but also Bruch or Prokoviev, Franck, Richard Strauss, Rachmaninov. I’d really miss their well-known works, but those are rare stand-outs in the entire oeuvre.
Stravinsky is a hard one to classify because he is unquestionably a great composer, but he impresses me more than that he touches me.
I’m not much of an opera fan, hence cannot fairly rate Bizet and Verdi.
So I’d get to
Tier 1: Bach, Mozart, Beethoven, Brahms
Tier 2: Mahler, Shostakovich, Vivaldi, Handel, Schubert, Tchaikovsky
Tier 3: Dvorak, Grieg, Schumann, Bruckner, Elgar, Bruch, Prokoviev, Franck, Richard Strauss, Rachmaninov and many more.
I’m not sure who I should put in Tier 4. Possibly I should differentiate and only put in Tier 3 composers who have shown good workmanship generally, which probably would move Rachmaninov down a tier. Saint-Saens would for me be in Tier 3.
Compared to the OPs list I’d remove Vaughn-Williams and Liszt who I wouldn’t miss at all.
Looking at the other posts: I’m not sure about the general output of a lot of early composers and therefore hesitate where to put them. I know only Pergolesi’s Stabat Mater which of course is as close to heaven as you can get, but as I don’t know his other work I’d hesitate where to put him. A lot of the late romantic and modern composers mentioned by the others would be Tier 3 or 4 to me.