The flu shot doesn't cure the flu - it's a vaccine that is only partially effective and only if you get it before infection.
Furthermore, if you went to your GP with the flu, do you know what he would recommend 99 to 100% of the time? Bed rest, fluids, and maybe some acetaminophen to bring down a fever. Know what contains acetaminophen? Nyquil.
The fact that your cousin got a very bad case of the flu doesn't mean anything about whether it would have turned out differently had he gone to a doctor when he first got the symptoms.