How does society decide what styles of argument are valid or invalid for practical purposes?
If valid here means logically valid, then it is not society which decides what arguments are valid, just as it is not society which decides that a tree looks like a tree.
Why are most people actually convinced that a method of inference like modus ponens actually work?
Most people do not understand formal logic and most specialists of formal logic do not understand human logic.
The Modus ponens is essentially the notion of general rule and everyone understands that if it is true that all bats are mammals, then any particular bat is a mammal. This really all there is to it.
All valid arguments have a logic and the logic of each valid argument is a logical truth, which I define as a self-evident logical expression, such as the Modus ponens, the Modus Tollens, the Hypothetical syllogism etc. Nobody decides that these logical truths are true. It is just self-evident. You no more decide that a Modus ponens is true than you decide of what you see. Whatever you see is by definition self-evident, but it is not you who decide what it is.
Since most people, at least people in good mental health, agree that the Modus ponens is self-evident, the Modus ponens has the same epistemological status as an objective fact, and society does not decide of objective facts.
Formal logic did not invent correct reasoning. Rather, humans recognise a correct reasoning when they see one, just like humans recognise light from darkness.
The difficulty, therefore, is not so much to reason correctly but to articulate some formal model of how human logic really works. Logicians have only been able to agree on a rather small number of rather simple logical expressions, such as the Modus ponens, and there is still no consensus on a formal model of human logic.
This absence of a formal model shows that humans do not need to be taught formal logic to be able to reason logically. This has long been known. Plato himself observed that uneducated people were able to give logical answers.
Logical reasoning is convincing because humans are naturally logical. All humans decide that a cat is a cat because it looks like a cat, which is just an application of a general rule, which is itself essentially a Modus ponens.
We are convinced because we are innately logical.