I'm here!
I think the main issue with the half-brother/sister issue with regards to the sire is that, as has been said, many stallions produce so many foals that it would be easy to claim a horse was closely related to a top-class horse. For example you could imagine an unscrupulous stud manager advertising a "half-brother to Galileo" at their stud, when in reality all they have is one of many sons of Sadler's Wells.
You also often hear about "full/half sibling in blood". This is generally when a mare visits two full brothers at stud, or a stallion covers two full-sisters. Whilst they are not 100% related, the bloodline has stayed the same.
I think the main issue with the half-brother/sister issue with regards to the sire is that, as has been said, many stallions produce so many foals that it would be easy to claim a horse was closely related to a top-class horse. For example you could imagine an unscrupulous stud manager advertising a "half-brother to Galileo" at their stud, when in reality all they have is one of many sons of Sadler's Wells.
You also often hear about "full/half sibling in blood". This is generally when a mare visits two full brothers at stud, or a stallion covers two full-sisters. Whilst they are not 100% related, the bloodline has stayed the same.

Then bollocks, Oddy's spotted it