[Kreisler jr]

I don't think Tchaikovsky is regarded as a second rate composer. However, there might be some discrepancy between his extraordinary popularity (at least for some of his works) and appreciation more based on "nuts-and-bolts" technical aspects of music or historical influence. Tchaikovsky is the most famous composer of ballets (only Stravinsky could come close and he is not as popular), his b flat minor piano concerto might be the most famous classical concerto of all, certainly one of the 3-5 most popular, and his violin concerto is probably also in the top 5 of popular violin concerti. There is more competition among symphonies but again his 4-6 are very popular and the same holds for some orchestral pieces.
Eugen Onegin is not as popular as some Italian Opera but still a candidate for the most popular slavic/Russian opera. Compare this with most other 19th century composers and few will have such popularity in so many different genres. Most had a narrower focus (e.g. no operas by Brahms, Liszt, Grieg, Chopin..., very little purely instrumental music by Verdi or Wagner and so on), only Dvorak comes close of contemporaries and he has only one or two superhits.

For some listeners (myself included) a few pieces by Tchaikovsky were a gateway into classical music but later have lost (some or most of) their appeal, thus the idea he is a "superficially emotional" composer.
It's hard to deny that a lot of his music is highly emotional or immediately appealing through melodies and orchestral color but there's nothing wrong with that. For me, this works best in ballett and "lighter works" (I have only superficial knowledge of the two famous operas and none of the others) and does wear off with repeated listening to some extent. He is not a great favorite of mine but I regard him as a very good composers with an extraordinary range (for some reason in the last decades "niche composers" like Bruckner and Mahler who excelled mostly in one or at best two genres seem to have eclipsed Tchaikovsky and others) and some highly attractive works.

 

[Kreisler jr]

Dvorak is criminally underrated. Listen to his Stabat Mater, St. Ludmilla, String Quartet no. 13, Hussite Overture, String Quintet no. 3, Piano Trio no. 3, Symphony no. 5, etc., and you'd find lots of masterpieces beyond the few "superhits." He's definitely on the same level as Tchaikovsky. Perhaps he even approaches Brahms.

I had probably heard many of the pieces you name before you were born and personally prefer Dvorak's music (although he is also rather uneven). But this doesn't change the fact that Dvorak has nothing like the dominance in a genre like PIT has in Ballet and I doubt that "Rusalka" approaches the popularity of "Onegin". Dvorak clearly "wins" in chamber music but this is a rather niche genre not helping enough with overall popularity and while his 9th symphony and cello concerto come close, they cannot really challenge the overwhelming popularity of Tchaikovsky's 1st piano, violin concerto. The New world symphony is about even with PIT 4-6 but the other Dvorak symphonies, even 6-8 are not that well known.

In any case, Dvorak for some reason seems either too likeable or not as overwhelmingly popular to provoke the derogatory comments that were heaped on Tchaikovsky in some quarters. Unlike R.Strauss, Sibelius, Rachmaninoff and Tchaikovsky Dvorak doesn't seem a composer a certain type of modernist commentators (like Adorno and Leibowitz) loved to hate (but maybe I just don't know, I hardly recall having seen any polemics vs. Dvorak).

 

[Kreisler jr]

Richard Strauss of all composers was first and foremost in it for the money. That doesn't mean that Strauss didn't have imagination or skill. Strauss' tone poems have a lot of color and vivid imagery. Also Sprach Zarathustra is a loud, rambling, fun-spectacular, Ein Heldenlieben, Alpine Symphony, Till Eulenspiegel, Death and Transfiguration, and Don Quixote the same; but I don't hear anything really heart-felt and sincere in Strauss

Strauss was a fin de siècle Nietzschean in several ways. He had ubermensch-like musical capabilities and used them ruthlessly (ruthlessness being another ubermensch-quality) which included making lots of fame and money with his music. I don't know if he personally also had the Nietzschean scorn for the traditional notion of "depth" although the stark emotions of Salome and Elektra show that it was not all only colorful rambling fun-spectacular surface.

[Tchaikovsky's] music also has a feeling that I find to be very "Russian", and filled with a peculiar kind of sadness and soulfulness.

Interestingly, some of his contemporary Russian critics found his music too westernized (which again hurt him deeply, I think).

 

[Kreisler jr]

I think that the universality of Tchaikovsky hides that he was at heart a "theatrical composer", A lot of his best instrumental music seems also informed by the dramatic or flamboyant gestures (say the openings of the 4th symphony or bflat minor concerto), great melodies and general graceful elegance of ballet. Balanchine even used the 3rd symphony as ballet music and I could imagine the string serenade, souvenir de florence and at least some movements of other symphonies, string quartets or the violin concerto in a similar way. Sure, the 6th symphony might also be in some way a model for Mahler (who reputedly appreciated Tchaikovsky more than was common in late 19th century Vienna) but the 2nd movement or the sentimental second subject of the first movement although sound almost like taken from ballet.
Not sure about Mahler but Bruckner, Brahms, also Schumann and Sibelius (although these two wrote theatre music) seem to be rather "anti-theatrical" in their better works.

 

[Kreisler jr]

Schumann's symphonies have probably never been as popular as Tchaikovsky's 4-6 but a lot of his piano music and piano concerto were and are core repertoire. In the last 25 years even Schumann's violin concerto (about the only piece where I find the cliché of the waning abilities of the physically and mentally ill composer understandable) seems to have gained some popularity with violinists.

 

[Kreisler jr]

To suggest Tchaikovsky was second rate is absurd. That means every other writer of classical ballet music is third rate -- since his are the best. That means every classical romantic violin concerto, of which his is among the best, is third rate. [...] It means every other romantic piano concerto, aside from his Concerto No. 1, is third rate.

I don't think Tchaikovsky was second rate but I don't think it is inconsistent to think so.
Lots of ballet music beyond Tchaikovsky, Stravinsky, Prokofiev is third rate. And so are many of the romantic concertos unearthed by hyperion or other labels.