The whole 'You can't have open borders with a welfare state' slogan is so confused on multiple levels. Now, obviously, taking the modal scope of 'can't' just to be within the bounds of the NAP, it is trivially true that the welfare state, in combination with open borders, is not permissible. However, that aggression is only found in the fact of the welfare state. The welfare state is the primary issue here, not the borders. It might even be the case that open borders would increase the amount of aggression done via the welfare state, but that does not make the borders the primary issue, precisely because open borders, as such, are not a crime, which is why bordertarians rely on talking about other aggression in relation to the borders question.
So when someone says, 'You can't have open borders with a welfare state,' all it is saying is that the welfare state is impermissible. That is obviously not in contention. However, this basic fact does not make open borders against the NAP.
For example, let me give an analogy to show how absurd the argument really is. Imagine a country is in a war, and if this country were to lose that war, it would get taken over by another country, and that country would have higher taxes than the former. Now, the only way for the current country to avoid being taken over is to conscript its citizens. If it doesn't, the country will be taken over. Clearly, the libertarian position is not going to be, 'You can't have no conscription with this war going on.
Just like no one would say that abolishing conscription in this case is a violation of the NAP, it is this also the case that open borders is not a violation of the NAP.