the best age to start playing chess



There's no "best age" because different people have different lives, personalities, learning styles and so forth. Like most things, being introduced when younger is usually a good thing, but it's not a requirement. I've known people age 65+ who took up chess and in even just a few years made lots of progress.
I knew of chess when younger, but hardly ever played. I didn't really start getting into chess until about high school to college age. My high school didn't have a chess team and I didn't play chess at all then, but around the time I graduated from high school is when I started getting more into chess.

Yes, there are some GMs who started after 13. This is extremely rare, certainly less than 1%.
So if you use this to prove that 13-15 is the optimal age, you'll have a hard time.