If a community persists in a practice that has known effects on future generations (and heartbreak for the generations bearing the babies) that practice can and should be questioned. And in fact, some in the UK in communities where cousin marriage was accepted are questioning it, while others are contemplating undergoing genetic testing as it becomes clear that the birth of babies with serious health problems can be avoided either by not marrying cousins or by doing IVF.
You're missing the point. As a previous poster mentioned, the majority of babies born with serious genetic conditions are born to parents who are unrelated.
Where genetic counselling is available, it should be encouraged for every couple if that's what they want, not just cousins. Targeting cousins is problematic for a number of reasons, not least that it gives some people the false impression that solely avoiding marrying a cousin is an effective way to reduce the risk of having a baby born with a genetic condition.
In some cases marrying a cousin would be less risky. For example, according to the NHS there are some ethnic groups who are at a higher risk of passing on genetic blood diseases such as sickle cell, due to the fact that carriers of the gene had resistance to malaria. So when there were malaria outbreaks in the past, many of those who died were not carriers, and many of those who survived were carriers, leading to a higher incidence of carriers in that group. So imagine a hypothetical case of a woman who is a carrier for sickle cell disease, and it runs in the family on her father's side. An unrelated man proposes to her, but her parents were hoping she would marry her maternal cousin.
If we follow the logic that marrying cousins should be avoided, she should accept the proposal (if she wants), and avoid the cousin. But what if the disease doesn't run in the maternal side of her family, and it turns out that the unrelated man is a carrier too? In that case she would have been better off marrying the cousin if her main concern is about passing on the disease.
In my area, testing for some genetic conditions is done as standard for certain ethnic groups as part of antenatal care (including white groups who have a higher risk of conditions like cystic fibrosis). This is too late in my opinion. What causes less heartbreak, breaking an engagement, or having to make difficult decisions about a pregnancy which is already in progress?
The other thing that has to be emphasised in all of this is that the language around genetic conditions has to be sensitive, otherwise it can give the impression that people with disabilities and serious health conditions are somehow 'undesirable', and that the parents of such children are 'selfish'. I was somewhat shocked at the way testing for Down's Syndrome was presented in antenatal literature, as if it was an automatic assumption that high risk parents would consider an abortion. I wondered how parents who already have a child with the condition would feel reading that during a second pregnancy.