Show winnings have nothing to do with the breeding LBs. They are based 75% on the mare or stallion's inherent strength (their ratings and how quickly/evenly they train/will train) and 25% on their offspring's strength. It would follow that a horse high on the breeding LBs/their offspring might do better in shows, but you don't have to show them at all for them to get up there.

COIs are something a horse's owner can choose to reveal or not because a lot of people were worried the COI update would devalue their stock (even though it really has no bearing on the game). If a stable chooses to reveal it, the information will be public, otherwise it'll just say unknown. Personally I'm not bothered by a degree of inbreeding in my stock, in part because I fell in love with KNN who can only breed to other KNN, so it's just about impossible to breed clean high rated lines, though I do kind of have my own line that's not as inbred as a lot of EEE+ KNN. I can definitely get having to shift a mindset from other games, though, and if you decide you want to do clean lines in a given breed I'm sure that could be a fun project.

The most sought after LB category would be Best Sires Eventing, which in all breeds is the ABLB that Deep Space Nine was talking about. It's generally the pinnacle of achievement to have boys high on there, though in the top 50 for your individual breed eventing LB is also very respectable. These are the boys who'll generally get used to breed for more high ratings horses. If you're breeding single discipline (SD) horses, then the prowess is similar for getting high on that discipline's all breed leaderboard or breed LBs, but comparatively few people breed for the SD LBs.

For AD (All Discipline) breeding, Best Sires Eventing and Best Mares Eventing are the two "ABLB". These are the top 100 strongest stallions and mares in the entire game.