Were they all commanders during WW1? If so, why didn't Britain, France and the US have similar staffs?
Defining "excellent" is a very difficult thing to do, but let me tell you why the Whermacht did so well in the early days of the war. They did do very well in the defensive also, but we will leave that aside for now.
Prussian principle of Auftragstaktik (spelling would be wrong) gave the German army and leaders a huge advantage over the allied armies that followed Befehlstaktik (Detailed orders). What Auftragstaktik was is very simple - Mission Command. The subordinate units would be given a broad plan and an objective they had to take and by when. The subordinate units had the flexibility to come up with their own approach and plans.
The merits of this system is very visible in the Battle of Eben Emael when the sheer initiative of First Lt Witzig & Sargent Meir (a junior officer and an NCO) enabled the Germans to take fortress on the Meuse. The task force under Witzig was simply told that they had to take a fortress with 80 soldiers. The rest (there was a plan, but it went awry) was left up to junior commanders. This was evident in the way the Wehrmacht handled Case Yellow (invasion of France) also. While there was a detailed plan, units assigned to certain sectors, logistics drawn up. Once they hit the border, this is what Guderian (commander of a Panzer Korps) told his troops, "The tickets they had been issued was for the last station, they could halt only there". The last station was the respective target on the channel coast. Subordinate commands responded to local events and decided on the best course of action.
Reorganisation by Hans Von Seckt (spelling again) - This was crucial for the quality of German commanders in WW2. Von Seckt early on (sometime in the early 20's) realised that static warfare was a thing of the past (at the same time, the French were building Maginot - that monument to static warfare) and the huge conscripted armies of WW1 would also not be sustainable.
He called the future very accurately. He predicted that the next war would be highly mobile, smaller (relatively) armies where training and doctrine would make the difference.
To staff the inter war armies (before Hitler took over), he and his core team handpicked some 3,500 generals from the Officer corps who had fought in WW1. They fixated on the nobility (better education, military tradition etc) and staffed the new Reichsheer with experienced officers who had a military lineage (of sorts). He also promoted some 500 talented, capable NCO's into commissioned roles.
Strict training regimen in the Reichsheer - The inter war army under Seckt had a very strict selection criterion, stricter exam norms, and almost impossible criterion if you wanted to be a General Staff Officer. Only the top 10% of the leadership was considered fit to enter the General Staff, of this only 1/4th passed the exams and entered the General Staff, so in an army with about 10,000 leaders (as an example), only 1,000 were considered even fit to enter the Officer Corps, of these 1,000 only 250 made the grade.
As you can see, it was a very potent mix. A very liberating and radical (not new, as it had been around since Moltke) doctrine, a very professional, highly skilled, highly trained core of officers around which the Wehrmacht could form and expand and the vision of the individual commanders themselves.
Movement vs Static warfare - Here was another area the Wehrmacht excelled in. A lot of German commanders (Manstein - the finest general of the war and top 10 all time great commanders, Guderian, Rommel, Model, Wittingstein, Balck, Student etc) saw the advantages and benefits in running deep operations. Interestingly enough the theory of deep battle was proposed by a Russian, Michael Tuchakvesky (sp?) and taken up by British theorists - J F C Fuller and B H Liddel Hart, but the concept of armored warfare where tanks acted independently of infantry was ditched by the Russian army (quite literally, the entire high command was purged by Stalin) and the allied armies.
The above two married each other perfectly - Mission command, fast moving Panzer columns staying in touch with via radio and resulted in some spectacular victories.
Compare this to the allied commands - every withdrawal, counter attack had to go up the chain and approvals needed for even tiny movements.
Most of the WW2 commanders had seen service during WW1, so there is no differentiation there.
At a tactical level, the German army was unmatched, but it was at the strategic level that wars are won or lost, and Hitler and his OKW / OKH lost the plot there.
A Good write up on Mission command type tactics
Some good sources are,
The Treaty of Versailles included many regulations for the German army one of them were a caped size. The Reichswehr ( name of German armed forces before Wehrmacht ) was limited to an army of 100.000 man. This had consequences for the average quality. The German army had one of the most experienced armies at the end of the First World War the reasons for that are rather obvious. The German army was big and had a lot of combat, more than any other nation, this created experienced officers. The German army was also pretty progressed at the end of the war. The reduction of the army to 100.000 increased the average quality of the army since only the best stayed in the army and only the best could join. Well known Generals like Rommel already had a lot of combat experience and had proven themself. The US forces for example had limited combat if you compare that to Germany so the officers of their World War 2 forces had naturally less combat experience. The highly condensed Reichswehr had thus a very high quality officer corps. Obvioulsy there are other reasons but this is one reason which is in stark contrast to other armies.
In almost every single piece of literature he's written, Robert Citino states that owing to their Prussian heritage, German officers and NCO's tended to be more aggressive and more proactive than their Western ally counterparts. This does not translate to the average German officer being qualitatively "better" than their equivalents amongst the Western allies - yes, German officers accomplished some incredible feats of bewegungskrieg by aggressively pursuing temporary and local opportunities, but German officers also committed some grievous errors by doing the same thing.
For instance, an aggressive, multi-pronged attack at the seam between American forces on the beach was the traditional, doctrinal response to their landing at Salerno, but Gen. Paul Conrath's efforts did not have an answer for the hundreds of 6" shells available offshore on the USS Boise and USS Savannah. Conrath wrecked key elements of his Italian Livorno and German Hermann Goring divisions for zero gain.
Similarly, the pre-war doctrine of Western allies was suited for some operations and not others. Commonwealth formations that amounted to the World War II equivalent of infantry squares were horrifically useless against the Germans and Italians in North Africa. American broad-front offenses were ineffective at breaking out of the beachhead at Normandy. Examples abound from all nationalities of both brilliance and incompetence.
As mentioned earlier, the limiting on the size of the army had some impact on this. First it meant that only having limited slots meant that only the best officers got promoted. I don't recall where I remember another tidbit from, but I seem to recall something about the restriction saying anything about the officer/soldier ratio, so they also maintained more officers than actually needed, allowing for a more rapid ramp up of forces later. Quite a few of the more promising officers also spent some time "observing" the Spanish Civil War and came back with some interesting doctrine ideas, including some of the blitzkrieg ideas out of it.
I think another part that can't be underestimated is some of the non-combat logistical management experience that the officers received during the various non-combat operations during the run up to the war. Starting with the German military rattling sabres along the French border, the annexation of Austria, and the take over of Czechoslovakia all required co-ordination in increasing scales of the various logistical ends of the military. The ability to rapidly re-supply mobile forces was something they then unleashed on Poland, France, and Belgium.
think this non-combat role and experience should not be underestimated as later in the war when their logistical advantage disappeared on the Eastern front, they suffered terrible losses and in the major counterattack in the West (Battle of the Bulge) one of the key items that ended the counterattack was their inability to re-supply the salient they created.