The biggest flaws with the film still has to be Red Skull. he just isn't fleshed out enough to be anything more than a mustache twirling type villain. Also are they even fighting Nazis? All we ever see is HYDRA. Did the Third Reich just completely fall in 1942 to HYDRA? A little confusing really. I personally interpret Schmidts little rebellion against Hitler as just a way to bargain for more independence in his actions but that he basically just still fought on the Nazis side, but it really isn't clear.
This is in line with the comics: Schmidt considers himself to be superior to the Nazis as the highest form of human life, and Hitler was afraid of him. So HYDRA has the same goals as the Nazis, but think they are the better people than the Aryan race.
Cap was part of the Proto-SHIELD group that fought against HYDRA, but Cap DID fight in normal WWII battles against the Axis powers too (they just didn't have time to show us these battles without disrupting the narrative). The movie takes place over a few years (I think), it just goes so fast you'd think it was just months.