You CAN write something better, hell I'd be willing to bet you could IMPROV something better on the spot and without notes. Not only are they shite, and yes, all of them are shite but HOTDQ is universally considered the worst one so far. While 5e is very good, and I think PHB and DMG are worth every penny the published adventures are absolute shite. Don't waste money on Horde Of The Dragon Queen.
Really though, you can play with just the Monster Manual and the free PDFs, so no it's not necessary. The Dungeon Masters Guide is best if you want long campaigns as it fleshes out the game, I also feel the variant rules in it are mandatory for certain playstyles. The Monster Manual is advised as it gives you actual options for encounters and if you have the Players Handbook, that'll give the players far more options. You can run perfectly decent games using nothing but the free PDFs on the website. For that and the DMG get them if you can afford them and want them. The proper MM adds a ton of flavor, but isn't necessary.
For monsters, you'll want to download and print the additional DM supplements for the adventures from WotC's website, which contain any monsters in the Monster Manual but not in the free DM Basic Rules pdf. In any case, you don't need the DMG or even the Monster Manual to run these.
Princes of the Apocalypse will be more costly, but it contains the equivalent of both Horde of the Dragon Queen and Rise of Tiamat (which was better received than Horde). If future adventures are of this caliber, then 5e will continue to do great. On the other hand, I strongly recommend Princes of the Apocalypse, which has been tremendously well received, and has even been called the "killer app" for 5e, meaning that this adventure really carries the flag for the edition. Horde of the Dragon Queen was not terribly well received, though I don't believe the people who say its awful - I think they're being rather hyperbolic, but there are valid criticisms. That said, it is my favorite of the three books. In fact, everything not in the free PDFs is really optional, but this is most true of the DMG.
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