Well I think that the 'modern' aspects of Mahler's music are quite important in assessing his significance and influence.
Having said that, I find the supposed link between Mahler's music and the Second Viennese School to be highly flattering to the latter group of composers. Similar in a way to the 'Brahms the progressive' thesis. Composers like Schoenberg felt like they had to keep on making the point that they were continuing some sort of tradition, as if to attach greater importance to their own music. For me, Mahler's mastery of form, expression, orchestration, aesthetic, structure, etc. is on a completely higher level, irrespective of whether his 'successors' were writing tonal or 12-tone music.
The link is not at all incidental; the Second Viennese School were heavily linked to Mahler in technique and expression, and they were the only ones who actually appreciated Mahler's music at a time when the wider musical world thought of him as either an incomprehensible noisemaker or a creator of overblown bombastic monstrosities (to list only the views that didn't openly express anti-Semitism).
Schoenberg's Society for Private Musical Performances programmed Mahler often, and he was likewise to be a part of the famous Skandalkonzert of 1913 if Berg's songs hadn't been interrupted by rioting. Webern conducted a performance of Mahler's Sixth in the 1930s that was supposedly quite revelatory, and Schoenberg conducted the slow movement from the Second in a performance that survives in recording.
On top of that, one can point to all kinds of reflections of Mahler in the use of particular timbres in the Second Viennese School, from the guitar and mandolin in Webern's Five Pieces Op. 10 or Schoenberg's Serenade to the hammerblows of Berg's Three Pieces for Orchestra, aside from the general soloistic character mentioned above, which contemporary critics linked to Mahler even while he was still alive. Mahler himself was good friends with Schoenberg (although they clashed often, as one might expect from two such egos), and purchased a whole bunch of his paintings anonymously simply to support his struggling friend.
Naturally, the music of any composer is not important because of its links to earlier important music, and none of this music would have had the staying power it has if it weren't successful on its own merits.
I hadn't previously read much on the influence of the Second Viennese School on Shostakovich, something which doesn't strike me as particularly obvious. Any attempt to cast Shostakovich aside to any kind of 'periphery' presumably is largely down to his persistence with a tonal language deep into the 20th century, something which clearly wouldn't sit with Boulez and other 'modernists'.
I don't consider Shostakovich's music any more tonal than the later Schoenberg. While he uses a lot of triads and diatonic elements, his music doesn't employ functional harmony except for occasional effect (which one can say of the later Schoenberg just as easily).
Anyway, Shostakovich admired Berg his whole life and said so frequently, despite the official Soviet line that disparaged the 12-tone method as bourgeois formalism. One of his early commentaries has him listing Schoenberg as well amid his foremost influences:
Articles Shostakovich published in 1934 and 1935 cited Berg, Schoenberg, Krenek, Hindemith, "and especially Stravinsky" among his influences.
The lasting appeal and greatness of Shostakovich's music transcends any and all trivial debate.
I wasn't attempting to disparage Shostakovich. I was simply saying that he's not central to my personal view of the 20th century. Some of his music is quite good.
