This one really depends on how you were raised and where. People who don’t have hot water, my wife’s upbringing in Puerto Rico in the city no less, for example. They didn’t have hot water, so they just jumped right in.
Anyway, if someone’s parents were strict about wasting water. I could see this being how they just developed, and it never occurred to them until an outsider questioned it for them.
One of those ol’ Reddit stories about a guy who didn’t know that no one else wore ‘shower socks’. Turned out his mother was crazy protective and was traumatized by a loss of a family member due to slipping in the shower and braining themselves. So, she forced her kid to always wear ‘shower socks’. His first adult relationship finally told him that that was weird AF, and he asked Reddit about it to see if it was just him. Yep, just him.
I always wash my flip flops in the cold water while waiting for it to warm up. There's probably something productive that everyone can do with the cold water.