Oooh this is interesting indeed. It would have been a solid way of creating a very effective warrior bloodline, but the people in charge of the ring weren't that forward looking.

They just wanted some good show for the gamblers and it would have been too much work to raise these babies, also they wanted messy fights not experienced fighters, and since most of the time the opponent was killed, it would be a lot of preparation, raising a baby to send him probably to his death in the ring.

In another setting, a more "polished" one, they would have trained or even sold those pups to gamblers that would have raised/trained them for long term profit!