Do you put savings and necessary expenses aside when paid?
To each their own. If it works for you great. I can’t imagine sitting down and tracking whether my grocery expenses fluctuated from one month to the next, since it can’t impact my ability to pay bills and save. If I was worried about not having money for groceries I would set that money aside when I get paid.
I also have a family and am the sole income earner. I just set aside everybody's spending money / allowance and they can do with it as they wish.
Your system sounds simpler but more rigid than mine. In my system, everything is a negotiable trade off. There is not static bills and savings to set aside, there is a trade-off between those things and spending. If I want to spend more on groceries or socializing or vacations in some period of time, that is just a trade-off against all the other things in my budget. Maybe I'd rather find ways to lower my bills than my spending. Maybe I want to save less in some month so that I can spend more on something that will bring us more joy that month. It is easier to make these decisions on trade offs with good visibility into all spending and saving.
But I would also suggest that it seems like you're actually doing the thing you said you don't understand why people do. You have a budget, it just has a relatively small number of categories. But the answer to your question is that people do this for the same reason that you do your setting aside of money for bills, savings, and spending.
I also have a family and am the sole income earner. I just set aside everybody's spending money / allowance and they can do with it as they wish.