You need to parse the sentence a bit. "85 to 90% of cousin couples do not have affected children" does not mean that the odds of one child being born with a hereditary genetic defect is 15%. It means that, for the average family size of a first-cousin couple, the odds are 10-15% that at least one of the kids is affected.

So, let's conservatively say the average family size among those who marry first cousins is 3. The odds of at least one in those three kids having a genetic defect are stated to be 15%. So that means the odds of any individual kid whose parents are first cousins having a genetic defect are a bit under 5% (the odds of a given event happening at least once in three independent trials).

The odds will be substantially lower if that 15% figure were based on a larger family size than 3.

As a baseline, tn the UK, the odds in the overall UK population of a genetic defect occurring are around 2.55%.

So the risk is roughly double the baseline for any individual child. But the way the numbers are presented makes it seem misleadingly high and has led to predictable screeching from the usual quarters. There is also no measure of severity. For example, despite my parents being unrelated, I have a genetic defect that causes high cholesterol levels in my blood. However, it's cheaply treatable (woo hoo, statins!) so its impact on pubilc health is next to nil.

I'd favour banning marriages where the partners have first-cousin and closer degrees of consanguinity, but I also see the point of not catastrophising the actual impact.