I have had an epiphany. The logic of the Primacy of Magic camp holds water after all; they have just made one simple error: the true essence of D&D lies in the Primacy of Burgling.
For every edition, the one absolutely necessary member of an adventuring party has been the thief/rogue. Dungeon crawling has simply not been possible without one. You could scrape by without a warrior, or a mage, or a healer, but without a rogue, locks and traps and secret doors simply become impassable. In 1E and 2E, only thieves/rogues have access to the skill system which allows them to tackle these challenges. Even in 3E, which opened up the skill system to every class, the skills for disarming traps and picking locks were capped for non-rogues -- uniquely so, that's how important this primacy is. And it clearly isn't just an arbitrary mechanical limitation. The rogue requirement naturally goes back to the very roots of the genre in Tolkien's Hobbit, in which Thorin and Company go through considerable trouble to recruit a burglar for their cause.
But then along came 4E. And Thievery became merely another skill. Now anyone could open locks, disarm traps, and even pick pockets just as well as a rogue. Were that not enough, the skill challenge system encouraged players to overcome obstacles as a party, gaining successes without using the Thievery skill at all! And, lo and behold, 4E was a highly divisive edition frequently accused of being "not D&D". Now, this connection between 4E changing burgling and 4E getting hated on is just a correlation I'm observing. I'm not saying that the Primacy of Burgling is the essence of D&D, except for the times when that's exactly what I said. And I'll acknowledge that nobody in this thread actually on the "4E is not D&D" side has made a case for the Primacy of Burgling. But, as has been frequently been asserted already, the cases they do make are factually incorrect and/or don't make any sense, so that makes Primacy of Burgling the winner by default.
Now, some of you may observe that in 5E, just as in 4E, thieves' tools proficiency is available to any character. But if you think about it the way I do, 5E handled this proficiency in a different way that preserves the Primacy of Burglary rather than blowing up my theory. You see, the only backgrounds that offer thieves' tools proficiency are Criminal and Urchin. 5E backgrounds are effectively a low-key multiclassing system, so these can be thought of as rogue "level 0s". Thus, even though you don't actually need to be in the rogue class anymore, 5E is still saying that, in essence, in order to burgle you have to be a burglar. 5E also left the skill challenge system by the wayside; clearly the developers thought that contributed to the problem as well. So the Primacy of Burgling is alive and well in 5E, and 5E is received as being "D&D" again. More correlation!
There you have it. All the evidence for the Primacy of Burglary being the essence of D&D, nothing but cold, hard facts. If you have any questions about the case I made, feel free to ask them and I will be happy to explain to you what you're really thinking and why it supports this theory too.