It’s extremely conservative. No minimalism, no Webern, no Stockhausen or Boulez, only Max Davies and Simpson to represent a non-deceased Britain, nothing from Russia for the past quarter-century.
Have you had a chance to review the other time periods? They seem to list very many pieces, a bit too many even to be talking about “essentials”.
Andrew Lloyd Webber’s “Requiem.” It won a Grammy for Best Classic Work.
His cellist brother Julian has recorded lots of modern classical composers, including Edgar, Brittan, Delius, etc. Julian has a vocation for digging up unrecorded works and recording them.
No, they’re decomposing. 
Prokofiev: 3rd Piano Concerto, 5th Symphony, Peter and the Wolf, Romeo and Juliet, probably some other stuff. (In particular, if we want to include some 20th century solo piano music, one of his sonatas might be a good choice.)
Shostakovich needs to be represented, but I’m not sure what to single out.
Barber’s Adagio seems to be considered essential, though I personally am not blown away by it.
For Ives, I’d pick at least one of the symphonies, probably the 2nd or 3rd.
Stravinsky: The Firebird, The Rite of Spring, Petroushka
Should Rachmaninoff be on here? He’s certainly big and popular, but not important in the sense of breaking new ground.
Some essential composers straddled the 1900 line, like Mahler and Sibelius.
That’s because Enterprise supplied the wrong link. 
Classical.net has a separate page for 1945-present (what they call “modern”), but they admit there is much under-representation.
For the earlier period, my offerings for “essentials” would have to include at least one part of the “Roman Tryptich” of Ottorino Respighi, probably “Pines of Rome”. If we can include Richard Strauss, any of the major symphonic poems would count, but I’d recommend “Don Juan” or “Also Sprach Zarathustra”. I’m also quite partial to Nielsen, Vaughan-Williams, and Prokofiev, but I’m not sure how much any of them could be considered “essential”.
Aha - that’s a much better list!!
I’d say the biggest omissions from that list (names, not pieces) are Lachenmann and Ustvolskaya.
God, NOOO!! Dreadful creature,* dreadful* music. Winning a Grammy doesn’t help his case
I’d add Ralph Vaughan Williams’ Lark Ascending as well as
Ottorinio Respighi’s Pines of Rome
A second (or later) to Firebird and Rite of Spring, too.
I thought I’d add a little more here, since the topic seems to have otherwise been played out (no pun intended). When you get into still-living or recently deceased composers, we’re really skirting the issue of “essential”, given that we don’t know what sort of influence any of them will have for the next 50-100 years. That said, here’s what I came up with for a sampling of truly “contemporary” composers and whatever works I could remember.
[ul][li]Sir Malcolm Arnold (1921-), English/Irish/Scottish dance suites[/li][li]Ferde Grofé (1892-1972), “Grand Canyon Suite”[/li][li]Howard Hanson (1896-1981), Symphony #2, “Romantic”[/li][li]Jennifer Higdon (1962-), “blue cathedral”[/li][li]Joaquin Rodrigo (1901-1999), “Concerto de Aranjuez” and “Fantasy for a Gentleman”, both for Guitar and Orchestra[/li][li]Christopher Rouse (1949-)[/li][li]Bright Sheng (1955-), “China Dreams”[/li][li]Toru Takemitsu (1930-1996)[/li][li]Tan Dun (1957-), “Water Passion after St. Matthew”[/li][li]Michael Torke (1961-), “Bright Blue Music”[/li][li]Joan Tower (1938-), “Fanfares for the Uncommon Woman”[/li][/ul]
Finally, two of the most influential/important early 20th-Century composers not yet mentioned: Charles Ives (1874-1957) and Zoltán Kodály (1882-1967).
I want to go on record with IVES as perhaps the 20th centuries most ignored, but most important composer. Almost anything in his catalogue is needed, but especially Concord, Three Places in New England, and his Symphonies. I think His Songbook is also a very important work, but am unaware of any attempts to create a definitive recording of it.
Carter hasn’t been mentioned and while I find him variable in quality, I feel he should be represented – perhaps his solo Cello works recently recorded by Yoyo Ma? He has aged well I think.
Do we include Bernstein? I think he fails to get some of the love he deserves especially for the Chichester (sp?) Psalms and his First Symphony. He was admittedly lazy at times, but brilliant nonetheless. I especially like his Clarinet Sonata, though I would be hard pressed to call it must have. His Ballet Dybbuk is an incredible piece of work and I would easily include it.
I support the Stravinsky recommendations.
Copeland’s Piano Concerto and his “Chamber” Appalachian Spring are probably his truly best work (don’t listen to the full orchestra version of Appalachian Spring, it is simply put a travesty of the original chamber work)
I like Gershwin, but find his best work only so-so in terms of important must have music. If he must be represented, I think Porgy and Bess is his greatest offering with his Preludes being second, the Rhapsody in Blue Third or fourth battling it out with his less popular Concerto in F. American in Paris could go on my “never hear again list” because it is simply too cute and self-indulgent for me.
Boulez is probably needed – Hammer without a Master is what they beat us with while I was in school, but I believe he did better work later, but “Hammer” is still pretty cool stuff.
Corligno’s Ghost Waltzes I believe I really liked, but they haven’t received a lot of play, so I can’t be sure. What did they give him the Grawemeyer Award for?
Definitely Webern needs to be on the list as does Berio.
I don’t find Babbit at all compelling or important – mostly unlistenable and an interesting theory that doesn’t pan out as music.
Mihaud and others – don’t feel like dredging through stuff, but we are mostly on track here (except for the ALW bit – who MAY belong in the Broadway/Musical list, but isn’t an important composer in any other respect. I will grudgingly say that to a great extent Broadway and Musical theatre in general owes it’s current success to his having contributed to it – I don’t have to love it, but I can accept it).