Derech HaMelech ‘the only time the Torah mentions a birthday is in connection with Paroh’
Not strictly true. Its the only direct mention. But of course we know the famous midrash on homon (bang, bang) that he chose adar cos its the month that moshe rabeinu was niftar, but didnt realise that it was also the month that moshe was born in, meaning that birthdays have a special good mazel or koach.
And the Greek army (I think) used to use warriors on the front line whose birthday was that day cos they had extra mazel.
The gemara brings that someone on their birthday who visits a choleh will remove one sixtieth of the illness. Other ppl only take a 60th of a 60th then a 60th of THAT etc etc