That's pretty silly. In-story it fits perfectly to have Sam step in as Captain America. It's always been clear that the role of Captain America is considered important and symbolic, which is why others have stepped in when Steve is incapacitated or unavailable. Sam was the natural choice to take over, because he's been Steve's closest ally for a very long time, and is a highly-principled hero in his own right.
To address your title question: Nope. Steve Rogers would not be Cap. Depowering storylines for heroes are pretty common, and at some point writers would want to do one for Steve regardless of his race.