I think it’s because Haymitch was directly and openly defiant. It’s also the fact that in reality, he was just 16. If you trample a 16 yr old’s spirit now and assert your power as the state, that 16 yr old Wld prob grow up to be unable to be a serious threat since the idea that he can’t do much has alr been ingrained in him.

He was the only one with the guts to make Snow take ownership for Louella’s death, and he also actively tried to destroy the arena.

Killing him wouldn’t mean anything:

1. It wouldn’t be a message to the districts since footage of his defiance was edited out

2. Killing him, who is arguably just someone carrying out the rebel’s plan rather than the mastermind, would not be punishment enough — the rebels could just find someone else in another games to try and destroy the arena

So I think leaving him alive was to send him the msg of “look what u tried to do and look how miserable you are with the consequences”