While you can't budget your way out of being poor if you have a very low income you absolutely can keep yourself poor by not budgeting no matter how much you make
I don't know why you seem to take offense with a simple suggestion that will help reduce how much you spend
There are people who make six figures who are in debt because they overspend and food is often one of the biggest factors in that
Those people aren't "poor". They aren't worried about eating or staying warm.
"Poor" is when you are deciding between fixing the car you need for work and whether or not you will have electricity the last 4 days of the month. You don't fix that with "clever budgeting".
But yes obviously there are levels to it however regardless if you are buying overpriced fast food when you could be cooking at home for much cheaper that's not good for anybody especially if you don't make a lot of money... So what's your point other than trying to argue over the definition of poor?
If you don't have an emergency fund and your car breaks down and then you have to get into debt to fix it and then you have to spend more paying interest on that debt and you get stuck in a cycle... Compare that to reducing your expenses by not buying fast food and building up an emergency fund and not getting stuck in that situation to begin with
Because people make political decisions about programs that support the "poor" and the definition matters.
In particular, if a program supports the "poor" and winds up handing money to someone who is making $100K, there are a lot of people who will scream about that and attempt to cut off all support for all poor people.
This was the whole the point behind the racist "welfare queens" dog whistles, for example.