No, it doesn't assume that there isn't a problem in Western countries. It assumes that the problem is orders of magnitude higher in countries where misogyny is ingrained in the culture. An honor killing in North America becomes national headlines, as was recently the case in the Toronto area. Rape happens everywhere, but the numbers for South Asian and Middle Eastern countries dwarf anything we can imagine here. According to an NPR piece, something like a third of Afghan marriages result from a rape, when the choice then becomes marry your rapist or die. They are not equally at play. That doesn't mean that each individual rape that happens in the West is any less violent than any individual rape elsewhere, but it's very suspect to compare the two at the population level.