What AVR did you try Audyssey with? The SR876 only has Audyssey XT, which is inferior to XT32 and will obviously lose against Dirac. It also sounds like you haven't used MultEQ-X (the main topic of this thread), so is your experience really up to date to the current state of Audyssey, and is it a fair comparison vs. Dirac?
I had the SR876, and when its HDMI board failed, replaced it with an Integra DTR 70.4 (which is XT32) - results with both of these were identical...
When my 70.4's DSP board failed (bricking the unit... I am still trying to find a replacement DSP for it

) I pulled out the old 876, and used it for many months via SPDIF (bypass the HDMI) until the new generation AVR finally got delivered.
So yes, the 876 was XT, but I had already compared its response in my room, with the 70.4 (which is XT32) and found them to be the same. (perhaps according to Audyssey, the additional filters provided by XT32 weren't needed in my setup?)
Also worthy of note, my 876 and 70.4 did not have the option of using either the Smartphone app or the PC app for adjustment - so the only option on these was the Audyssey defaults.
These defaults are unchanged. - My gut feeling is that if there had been a way of disabling MRC, then I would have been fine with the Audyssey performance.... and would perhaps have preferred it to without RoomEQ. - But the Midrange is absolutely the heart of any listening... and the default Audyssey configuration, with MRC enabled, does nasty things in the midrange. The default Dirac setup, does not have this issue.
So yes - defenders of Audyssey will say that with the current generation of Audyssey apps and tuning (and the Denon/Marantz AVR's that support it) better results are possible.... Perhaps true - but I had the choice between taking another chance on Audyssey, after 2 generations of disappointment, or switching to Dirac, and the results with Dirac were (are) absolutely stunning.
Now if I was the owner of any of the multiple generations of Denon AVR's that support the MultEQ apps, I would definitely go down that path, and I would find it extremely difficult to justify the additional $$ to either purchase a Dirac Licence (for the current generation) or upgrade the AVR (for older generations)... I would lay bets, that you could get an Audyssey setup tuned to match a Dirac setup - or so close that Blind testing is unlikely to show a definitive winner (preferences yes, winner, probably not).
So yes D&M's move to enable optional Dirac is interesting, but I think it is mostly relevant for users of DLBC, and in the future DL-ART.
For the majority of users, who simply run the default and don't mess with complicated apps.... my experience would seem to show that Dirac does a better job. (well it did for my setup and my ears)