I'm not talking about the methodology; I'm talking about the results. The result of inbreeding is genetic insufficiency for survival - and there are a lot of human traits that have been preserved that are not fit for survival either. Yet, we don't forbid breeding for specific traits.
Generally humans and animals have different rights in terms of what they are allowed to decide wrt their breeding habits.
Additionally, humans FREQUENTLY choose to adopt rather than risk passing on a genetic abnormality to a biological child.
The only difference here is that we would be making the decision for the tigers since, as everyone would agree, they don't have the ability to understand the consequences and make the decision themselves.
As long as frequently is considerably less than 1% of all births in the US (and significantly less in other parts of the world).
1.5-2% of children are adopted in the US. About 50% of those children's parents adopted due to infertility, of which an even smaller fraction is voluntary infertility.
I wouldn't say that is something that humans due frequently, and is probably limited to a small set of cultures, and an even smaller subset of parents with both the knowledge and fortitude to make that decision.
I'm an adoptive parent and know several dozen other adoptive families. I have never personally met someone who chose to adopt to avoid the risk of passing on genetic abnormalities.
As an adoptive parent myself, I know several couples who chose to not pursue biological children because of genetic conditions.
Unfortunately, I also know some who chose to ignore the advice of their genetic counselor and now have 3 lovely, but severely mentally disabled children (Fragile X run amuck) who will never be able to live independently.