Midwives have been told about the benefits of “close relative marriage” in training documents that minimise the risks to couples’ children.
The documents claim “85 to 90 per cent of cousin couples do not have affected children” and warn staff that “close relative marriage is often stigmatised in England”, adding claims that “the associated genetic risks have been exaggerated”.
As you stated, the worst thing of such marriage is having kids with health problems that can accumulate very fast with each new generation(silent mutations that get only worse and someday pop out with loud bang). This is mostly the only thing that stops such relationships.