You will not find a zoo that crosses lions and tigers. You may find roadside menageries that do so, but no accredited zoo would purposely let that happen.
As far as fish... line breeding is a little different. With bettas and other line bred fish, you can clearly see issues that stem from inbreeding. Tail nipping, deformities, breeding issues, etc. Keepers have to outcross every 10 generations or so to keep everything from going completely goofy.
Hybrid fish are not kept by serious keepers. Flowerhorns, parrotfish, all that crap is shunned by actual hobbyists.
Snake breeders create new morphs, etc, from inbreeding, but don't use related animals if they can help it. Like fish hobbyists, many people don't like the hybrids.
I think the issue comes down to, as you will read across the forum, conservation of seemingly pure bloodlines. With the limited amount of importation, people would like to do their best with what they can.