Let’s define terms here. For want of better ideas, I’ll re-cast classical music as “symphonic”. This is the music that orchestras play from the repertoire of European traditional composers, based on forms laid down by the Church and then elite society from the 16th century to the early 20th century.
That allows me to use “classical” as “Classical”, the important and formative time of 18th century symphonic music. I put Hadyn, Handel, and Mozart here.
Bach is a bit of a precursor. He is the transition from earlier, Church-dominated forms to the secular forms of Classicism, but he’s also a brilliant and innovative composer.
Mozart dominates the classical period, followed closely by Hadyn. Mozart is so unusual in so many ways that you could never leave him out as one of the top people in the symphonic tradition. Yet Hadyn not Mozart really invented the symphonic form.
Handel brings the symphonic form into the English-speaking world.
Beethoven is transitional. He starts out in what we normally call the “classical” tradition, but at the end shows definite Romantic influences. He uses classical devices to express a Romantic lyricism. Overall, I would rank him below Mozart and Bach, though, and on a level with Hadyn and Handel.
If you take away Brahms, who do you leave in his place to represent mid-19th century symphonic music?! You can’t take away Brahms, dear sir!
Chopin is very important for the piano, but no more. He rates below Brahms overall.
Mahler, like Richard Strauss, is a bridge between the Romanticism of the 19th century and the much more complex and exploratory style of modern music.
I find it easier to mark the truly second-level composers than I do marking the first-level ones. Chopin, Liszt, Schumann, Mendelssohn, Saint-Saens, Bruckner, are definitely not at the top. Still, they’re well-known.
Tchiakovsky is at the second-level, though I admit he’s amazingly popular. I think Rimsky-Korsakov is more important.