Captain America was "made" sometime AFTER the war had started (around 1940) and "disappeared" sometime before the war was over (before 1945). Captain America was not around in WWII for very long.
The movie was simply following the comic books that it was based on. Captain America was only around for a few years before disappearing.
The Avengers started as a superhero team of Iron Man, Hulk, Thor, Ant-Man and Wasp. In issue #4 (link) the Avengers find Captain America and unfreeze him, and he joins the team. By issue #16, the founding members had left the Avengers, making Captain America the de facto leader (link) (and he chose interesting people to be Avengers: Scarlet Witch and Quicksilver (who were part of Magneto's bad guy team) and Hawkeye (who started as an Iron Man villian (link)
The Captain America movie had to end with him being frozen so he could appear in the same timeframe as Iron Man, Thor, Hulk, Hawkeye and Black Widow and join the Avengers!