The fully-rated DB 605A and 603A engines that will not be blowing themselves to pieces in a normal service use, were 1350 PS vs. 1625 at the altitude of 5.7 km. Or, 0.83 vs. 1, or indeed the 603A was making some 20% more than the 605A.
This 20% is, BTW, almost exactly what you'd expect from a simplistic comparison of the displacement and rpm: 44.52/35.7*2700/2800=1.20. More volume gives you more power, but lower max rpm for a given mean piston speed eats away some of that power increase.
20% is not magical, by far, but every little bit helps. Doesn't obviate the need for a better (2S?) supercharger for high altitude performance, of course. That being said, maybe 20% is small enough that one could argue they would have been better off working towards increasing the power output of the 605?
German fighters were much smaller than the best Allied stuff, though, that also can help with drag.
Not an expert, but... This engine looks VERY impressive on paper, however considering that the Italians then canceled this engine in favor of license producing the DB 605 suggests that it wasn't all plain sailing?
As far as I know, it was almost exclusively bureaucratic issues rather than any inherent problem with the engine itself.
Italy cancelled almost all liquid-cooled engine projects in 1933 and only restarted them in earnest by 1939. The AS.8 itself was originally made for the C.S.15 racer which was designed to retake the air speed record from Germany, but the secondary target was to make a high-performance fighter engine for the Regia Aeronautica - hence its much greater emphasis on reliability than its older brother AS.6.
Fiat was ordered to abandon the engine upon Italy's entry to the war but was told to restart it and make the engine an inverted V like the DB engines, resulting in the entire engine needing to be redesigned and creating the relatively underwhelming A.38. By the time they could get the A.38 into production it made less horsepower than the DB 605A which could be obtained in great numbers from Germany, so the project was cancelled. Any further attempts at introducing the AS.8, A.38, A.40 and A.44 were quashed by the armistice.
The AS.8 was basically ready to go by mid-1942 historically, without the hiccups it might've been ready as early as mid-1941.