Ha-ha, you are brave, congratulations! This sentence is presented in the medical chapter, it is meant to be used mainly from a medical perspective not in a usual conversation about your health. But your colleague is not right, the sentence is perfectly correct, but in a different context, for instance:
The colleague: Mă simt foarte rău. Mă doare capul.
Rapuchan: Ai o răceală? Sau ești doar obosit?
But indeed, if you start the conversation, observing his state, it is appropriate to start with: ”Ești răcit?”
Thanks. I just showed your response to my colleague. He agreed with you and added that either way "Ești răcit?" would still sound better and "Ai o răceală?", while being technically correct, sounds pretentious to him. So I guess this sentence should have come with a warning attached. :)