You narrow your odds from 1/5 or 2/5 to 1/4 / 2/4. Also the person making the kill is confirmed town so your chances are even higher than that because there is no anti-town swaying your decision (hopefully)
If you're hunter(or any Townie for that matter) you always have a 1-2/4 chance your death doesn't change the odds of you targeting scum.
You die and the person you attempted to Lynch dies with you be he a Wolf, a minion, a Tanner, of a Townie. Your odds never change as hunter.
As a Townie, lynching the wolf is basically just saying you'll add your Lynch to whoever he lynched, if he's not god damn sure who the wolf is there's no point in lynching him because he has just as Shitty a chance to hit target as anyone else. And if he does know who the wolf is he might as well say it in the thread and you can Lynch that person alongside him.
It takes at least two lynches to kill a player. If three players Lynch one person they will be the only kill off the night, but we can't even do that because there's a bodyguard in the game, and bodyguards can't Lynch. So the only way you could even get killed is if a Wolf lynches you also (which they won't because they know you're the hunter) or you Lynch yourself (which would make your hunter kill into a single body double murder suicide)
There is no good reason to reveal you're the hunter except to counterclaim an anti town that thinks it's a good idea to get lynched.