Actually, if we are to omit something, it would be the vermelha, not carne. We never actually say "carne vermelha", unless it's really necessary to specify that it's beef and not any other kind of meat.
Anyone saying just carne will be commonly understood to be referring to beef C:
Vermelho in this case would only be used alone if the question was which kind of meat they like/want, etc. Alone in a conversation without previous mention of meat, it would only mean the color red.
The working class people that I grew up with very rarely called beef red meat, just carne de vaca. Or músculo! I still remember being astonished when my friend ordered X kg of músculo at the meat counter, and she received ground beef (no fat:)! I thought it would have been called carne moída, which i believe can also be correct.