Busy has become go-to, normal and even glorified, because we use it as an excuse to avoid other things.
Constantly convincing ourselves how busy we are is a convenient way to avoid the things we don't want to push ourselves to do. Think about it: "I can't exercise because I'm just too busy," "I can't eat healthy because I have no time to cook!" "No sorry, grandma, I can't come to your birthday party because I am just way too busy."
We also use the "busy" explanation because our culture socializes us (and mothers in particular, I think) to feel a need to please, to be needed by others, and to feel important.
And especially given that we are living in the age of social media, our culture has become obsessed with status and competition. Busy can be part of that: it's a defense mechanism, a go-to "status symbol" to indicate your importance. By being busy all the time, we are trying to demonstrate to ourselves that we have full and worthy lives, that we are needed and