Airlines will always try to take steps to reduce the risk to a traveler, even if that sometimes seems contrary to the passenger's immediate interests. When an airline asks for a doctor's note or similar for what may appear to be a very minor issue, it is not because the airline wants to cause a hassle but rather because it is better to be safe than sorry.
I had a sad experience a few years ago when working for a previous airline employer. A lady had been diagnosed with terminal cancer while on a visit to the UK and had been deemed unfit to travel. Her family however wanted her to return to her home country to die rather than in the UK as she already held a return air ticket and they could not afford to ship the body back as cargo. As the airline, we required a doctor's note clearing her for travel. This was provided, and our staff even called the number on the doctor's letterhead to confirm that everything was in order and to verify any special handling instructions.
On the day of departure, the lady seemed a bit disoriented but keeping in mind the doctor's clearance and her husband's advice that she had taken a sedative, was cleared to travel. Once airborne, she was fine initially and even ate the first meal before going to sleep and covering herself with a blanket. Sometime during the night she had difficulty breathing and passed away. Her husband actually noticed her breathing difficulties but declined to notify the crew as he was afraid we would divert and cause additional costs to the family. When the crew came around to do compliance checks at top of descent, he calmly stated that she had died about 3 hours earlier.
On arrival, the matter was passed on to the local police who asked the husband to give a statement. While left unattended to complete his statement, he fled the police station. The body was subsequently claimed by the brother of the deceased. When attempting to contact the doctor named on the letterhead to advise of the death, it turns out that the letter was a forgery using a modified telephone number on a genuine NHS letterhead - set up by the husband.
Local police declined to prosecute as they claimed the matter was out of their jurisdiction as the death had occurred prior to entering their airspace. The file was passed to CPS in the UK but they also declined to prosecute as all parties were non-UK citizens/residents and the actual death occurred outside the UK. Basically the guy got away with murder.