Extinction actually has a specific meaning in behaviourism, it refers to extinguishing an unwanted behaviour (signalling at night) rather than wanting to extinguish the baby - it's a term used in behaviourism and can be applied to any unwanted behaviour, not just crying at night.
The connotations of using that term are rather unfortunate but there's a reason why that name is used.
The naming / branding of sleep training methods in parenting is abysmal. “Cry it out” is also called “extinction.”
Like who came up with this? It’s terrible, and totally reasonable to reject the notion based on the naming alone.
That said the right way works. Even if you know all of the technique and carry it forward faithfully it can take a lot of bravery to go ahead with it.
Anyone on the fence with this I get it. There’s a subreddit, /r/sleeptrain where you will find others in your spot.