Some questions to ponder:
What about the wing loading and turning radius?
It should be higher on the DIII due to smaller wing area, but wing loading alone doesn't determine turning radius, I believe, lift also must taken into account.
Wing loading indirectly tells us how dense is the airplane, wich tells us about its diving qualities. I figure out that having same weight, smaller wing area, and having less drag (smaller wing surface, wing mounted radiator instead of fuselage side mounted, the DIII would be faster in a dive, and accelerate faster initially.
And roll rates? I believe that the DIII, with less mass on the wings would roll better, but I have no clue if that's significant for a WWI airplane, or it only depends on the ailerons. Wich brings us to the next question.
What about the DIII ailerons? I think they are very stylish with that raked shape, but wonder how effective they were. On the contrary, the British seemed to think more was better and their fighters usually had two pairs of ailerons.
What about the lifting efficiency of the sesquiplane configuration? Less interference between wings has already been noted, but I wonder, does the bottom wing of the DIII qualify as one of those high aspect ratio wings (long and narrow) wich are very efficient in providing lift, or not even close?
In a nutshell, how good were they as turning fighters compared with each other and their opponents? What does the story of the duel between
Manfred von Richthofen in his DII and
Lanoe Hawker in his DH2 tell us? Yes, I know bringing up the performance of a pusher will elicit derision, but seriously, anachronic as they were, I have the feeling that those pushers, though slow, were quite agile and capable of quite tight turns.
How did the DII and DIII compare with their opponents, the Sopwith Pup, the Nieuport 17, the SPAD VII?
From McCudden opinion of a captured Albatros, it was hard on the controls, wich is not a desirable thing, and understandable in view of that large elevator and rudder, and the two hand grip on the control stick suggest that it recquired strength to pilot an Albatros. But on the other hand, that was a subjective opinion from somebody used to light and handy airplanes. McCudden went out saying that he couldn't figure out how the German pilots managed to handle their Albatros so skillfully, wich suggests the machines were quite maneuverable and not sluggish bricks with wings.
Yet on the other hand, the impression one gets from the analysis of the Albatros is that it was built with speed, power and climb in mind, not dogfighting. In effect it seems that
the Jasta pilots were using their advantages in speed, climb and ceiling for zoom and boom tactics (or energy fighting if you will) whereas British pilots influenced by their handy rotary engine fighters were still thinking of the turning fight, (the Sopwith Camel being the logical evolution), at least until the SE5 arrived. In my view , this is one of the things that explained the SPAD VII relative lack of success in British squadrons.
So sometimes , it seems as if the DIII was only a marginal improvement over the DII, and the German aerial supremacy in
Bloody April was just due to sound tactics, the simple fact the Germans had now a fighter with an engine powerful enough to carry two machineguns without performance loss, and last but not least, the woeful obsolescence of so many British types.
But then again, why did the Germans copy the Nieuport 17 wing? They couldn't play at the Allies game of light, nimble rotary scouts, and if speed, power and climb were the way to go, the DII should have been good enough. Was the adoption of the sesquiplane concept a step in the wrong direction? Hard to believe. Exactly, what made the Nieuport 17 so feared and admired by the Germans? Well, on one hand when it appeared it was so revolutionary that it surely made the German pilots flying in Eindeckers feel an inferiority complex similar to the one later felt by British fliers in pushers when faced with the Albatros.
One reason that has been pointed is improved visibility downwards. Certainly it is very important, but situational awareness comes second to performance, I think. Else, you end up with a failure like the Airco DH5, the view from the cockpit was wonderful, too bad the only thing you saw was the undersides of the enemy fighters overhead

(ok ok, i am digressing and stretching it, but you get the point). I think that in order to compare the DII with the DIII, we have to begin examining what made the Nieuport 17 so good, and how was that applied to the DIII.
So I think the qualities the Germans liked in the Ni 17 and wanted to emulate were 1) visibility 2) climb 3) perhaps good turning ability and/or roll rate in spite of the decreased wing area?
That's the impression I am getting from the responses in this thread. Else, why bother with the sesquiplane concept at all? I don't think it's a coincidence that it was adopted in the last biplanes of the 1930s, including fighters renowned by the agility such as the Fiat CR32 and the Polikarpov I-15. Surely the designers must have been onto something.
I am believing now that the DIII could do everything the DII did, only better.
I hope I have managed to say something that is not wrong and makes sense in my long ramblings

I have just been wondering about the true qualities of the Albatros fighters for years.