Well, no list is going to please everyone. I'm certainly not a fan of that one. Also, I'm not really sure what the criteria are. Is it favorite composers, or most "important" composers, or some mixture of the two? I actually have objections to the majority of this list, if we are talking about "importance", which is the only thing that can really be dissected beyond personal taste:
1. J.S. Bach
Bach is surely not as "important" as Handel, considering his work would have been unknown to a huge number of other "important" composers, whereas Handel's would be well-known. I do prefer Bach, but Handel was much more influential during the period in which music was actually still being influenced, to a great degree, by Baroque music.
2. Bela Bartok
Bartok is an iconoclast, and his music is far more one-dimensional than Stravinsky's, and far less influential. Choosing Bartok en lieu of Stravinsky is so illogical that I can't possibly think it's anything other than an oversight. Stravinsky, Dufourt, Cage, Xenakis and Schoenberg (and Ives and Babbitt and Reich and Webern and Stockhausen etc. etc. etc.) are almost certainly the most influential composers of the 20th century, so to include Bartok and exclude four of those five is criminal.
3. Ludwig van Beethoven
This I can't argue very strongly against, although I would say that Schubert, if we're talking solely about innovation, deserves the spot, if only one of them can have it.
4. Frederic Chopin
Liszt is probably the most influential composer of the Romantic Era, so to choose between Chopin and Liszt makes me think again it would have to be sheer oversight to select Chopin. Again, I am not commenting on quality of music, but Liszt is extremely important. Chopin was just good at what he did.
5. Claude Debussy
I would say this is correct, if we must select between Ravel, Faure and Debussy.
6. F.J. Haydn
I would say this is correct in a way, and completely backwards in another. Haydn shaped the Classical era, but Mozart shaped everything else.
7. Gustav Mahler
This name has absolutely no right to a list this exclusive. If anything, Mahler should be on a reverse of this list, as he taught kids it was alright to take a step backward. By far the worst inclusion.
8. Olivier Messiaen
Messiaen's role in music is extremely complex, and I have no doubt in my mind that the person who compiled this list does not understand it. In that sense, I suppose my objection to this name being included is more epistemological than factual. What is almost certainly Messiaen's great legacy is his list of students and championing of then-contemporary music, not his actual music, in the sense of how much he changed the landscape of music to come. Even his sparse list of "innovations" is vastly overstated.
9. Arnold Schoenberg
This can't really be argued with. However, Stravinsky deserves this spot more than Schoenberg.
10. Richard Wagner
This one doesn't even make sense to me. I am unsure as to why he's on this list, so I can't argue with why that reasoning is stupid. If we're talking about Expressionism, motivic writing and harmonic innovations, we should be talking about Berg and Scriabin, not Wagner.