One does not imply the other.
Really? The saying certainly seems to indicate that using the threat of violence to enforce societal norms is a good and desired thing.
What it implies is that there are a number of people who would do bad things without the deterrent of firearms.
Does it? The saying is not "an armed society is a crime free society".
There are many reasons besides fear to support deterrents to crime.
Sure, but the first criteria should be effectiveness
An "armed society" is not the reason most people are polite to each other
But that is exactly what the saying says: "an armed society is a polite society". The clear implication is that an unarmed society would not be polite, which is the opposite of what you are saying.
I would posit that most criminals are cowards, with the exception of those with a predilection to violent crime.
I would question that assumption. It seems to me that criminals tend to take more risks than the average person. That means their either less cowardly or more desperate. In which case, in a society where extreme violence is used to enforce society's rules they may well have an advantage over their non-criminal companions.
Frankly, I think many people need to think about that saying a bit more. Why is the armed society polite? Because rude people are killed. If that's true, then it must follow that violent people are in charge of enforcing the rules. So what about the people who allow the violent people to enforce "politeness"? Are they ok with people be killed because they were rude and thus callously indifferent? Or are they too afraid of the violent people to object? After all, objecting to your violent benefactors, might be considered rude and we know what'll happen then...
It seems to me that in any situation where "an armed society is a polite society" were true, it would effetively be the same as "might makes right". So when might makes right, who ends up on top of the heap then? It seems to me it's those with a predilection towards violent crime.