Are the other militaries of NATO in awe of the US military when the US military shows up for training with them?The simple answer is no. Professional soldiers generally have a good idea of the size of foreign military budgets and the condition of their active forces. Therefore, it is no great surprise among the Germans, the Dutch, the British, or the Canadians to see U.S. Army or Marines deploy with more vehicles and helicopters than their own forces can muster. What is of professional interest are improvements in the quality and capabilities of new weapons and equipment - and what these capabilities mean to the soldiers who do the fighting. I can say with certainty that no NATO force is interested in the M-16 rifle or any of its variants. It is not a weapon that European soldiers admire or feel the need to copy. That was not true, though, of the old M-1911A1 .45 caliber Colt automatic pistol. Almost all European automatics were based on this design but used smaller caliber bullets (mostly 9mm). NATO troops loved the .45 pistol and enjoyed shooting it when they could get their hands on one. Conversely, U.S. troops loved some of the weapons used by our NATO counterparts - especially their light machineguns and some of their rifles which are first rate. Overall, some foreign equipment is going to be superior to items used by the American military establishment and some U.S. equipment (the Blackhawk helicopter in its heyday) is going to be the best of it's kind.
From a personnel perspective - recruitment standards from one nation to another vary based on several factors with factor # 1 being the popularity of military service. If the military is viewed with admiration and its members reflect that esteem - then many talented, able people will present themselves for service and the pool of soldiers and sailors will be of good quality. On the other hand, if the popularity of the military is declining - it can be difficult to attract enough solid young people for service. The U.S. has been able to attract enough high quality young people to meet its mission - despite periodic lapses and occasional shortfalls. Consequently, high skill positions are filled competitively by volunteers who are highly motivated and highly coachable. This may not be the case in every NATO army. One would need to have access to information concerning the achievement of quality goals among each year's cohort of new recruits to make meaningful comparisons.
Every army has its own internal "elite" formation. The British have the SAS, the German Bundeswehr maintains exceptional infantry and armored units, the French have the Foreign Legion and their Paratroops. These are special units and are not directly comparable to each other. However, except for a "Foreign Legion", the U.S. has some kind of counterpart to each of the elite formations mentioned above. There can be considerable professional admiration and respect for a foreign military without the sense of "awe" that the question invokes. Soldiers' work is virtually the same from one army to the next - the language and technique may differ but the basic military culture is universal.