Well the roads are usually provided by private enterprise, though paid for by the state. But it is still ironic because the argument is not no government vs. big government, it's small government vs big government. Citing the roads is not a point in support of getting the government more involved in public life, because if maintaining the roads and other things like that were the only actions required by government, then the government would be miniscule.
What would not be ironic would be something like 'ironically, the defenders of government always mention the military as their first example - I mean come on people, have some imagination at least!' That would just be frustration.
Put simply, if there's a range of possible examples people can pick to support a given stance, and they always seem to pick one of the weakest/least-appropriate ones, then yeah, that's ironic. It's a state of affairs or an event that seems deliberately contrary to what one expects.
I think you're just trying to salvage an incorrect application of the term, and I don't think we're going to get much further here.
As an aside to that, your last phrase a state of affairs or an event that seems deliberately contrary to what one expects highlights a significant difference between British and American irony. In the US, you can choose to wear an ironic t-shirt. With British irony, you can't - irony is something that is unintentional (it can be known, just not intended). As soon as you intend to do it in any way, it's deliberate or expected and so becomes unironic.
I've heard British people claim that American's don't 'get' irony but i've never heard it claimed that there's a difference between 'American and British irony.'
In my experience the irony police are usually unwilling to consider the full subtext of someone's statement when they describe it as ironic. For instance people often say things like 'ironic, isn't it - beautiful weather all week and then when we finally decide to throw a barbecue, it rains. ' This is of course just bad luck, but people like to jokingly invoke some sense of fate conspiring against them. Or, someone might consciously wear an 'ironic t-shirt', and people consciously recognise the intent, but the unspoken subtext that maintains the irony is that there is some other observer or potential wearer who would be unaware of the incongruity.
What would not be ironic would be something like 'ironically, the defenders of government always mention the military as their first example - I mean come on people, have some imagination at least!' That would just be frustration.
Put simply, if there's a range of possible examples people can pick to support a given stance, and they always seem to pick one of the weakest/least-appropriate ones, then yeah, that's ironic. It's a state of affairs or an event that seems deliberately contrary to what one expects.