Shall I accept diagnosis given by GP that the pink phlegm after bad coughing is due to grazing of throat, without them checking my throat?

I have been coughing for 3 months. For last 3 years, I have been coughing like this so normally I assume I need to be healthier. However, this time I have noticed SOMETIMES my phlegm that I cough out is pink in colour.

I have been to the doctors twice, and both keeps dismissing the bloody phlegm as a result of heavy coughing grazing of the throat walls WITHOUT checking my throat or doing examination. The phlegm is now thicker and still pink to red. I asked for a chest x ray because my chest is paining.