Is that definition not supporting your points though? It defines gender as mostly a social construct, which imo reinforces the fact that it’s made up and not a tangible thing anyway.
Sometimes biological sex matters (e.g. as medical info for a doctor to understand) but other than that it’s connected to gender in name only, based on made-up social rules.
If he was counting his money in $100 bills it would still take him about 40 years,
Edit: assuming he counts 1 $100 bill per second