I only stress this because I've seen these same questions asked over and over again for years. If you ever have any thought-provoking questions about the game mechanics and seek clarification,
wiki is a great place to start.
-As others have mentioned, killshot doesn't matter
-Enmity levels make no difference
-TH does not have any effect on treasure coffers
-TH has an effect on 'rare' & 'r/ex' drops
-& much much more
To expand on what Viewpoint was saying, here is concisely what we know about activating TH:
"Treasure hunter 'comes into play' when:
(1)The thief (or /thf) produces enmity on the mob in question
(2)The thief performs an action on that target mob"
Pretty much ver batim from SE, though it is unknown if conditions (1)&(2) are supposed to be concurrent or can be separate, but my guess is concurrent (i.e. engaging mid fight, missing an attack (0 enmity), then /whm curing an ally, and disengaging and afk'ing, does not activate TH).
Viewpoint takes it one step further and mentions that the thief could die, HP, change jobs, drop part, and log off, and TH would remain unless the mob cleared its hate list and regened a tick. He's most likely right. I won't say 100% due to some of the vague words SE uses (i.e. 'comes into play' as opposed to 'TH is incorporated into the calculation of drops at time X'). Better safe than sorry, but he's probably right.
* * *
There are a lot of good facts in that wiki article. But one I'd like to bring up briefly before finishing, comes from a really old interview. Like 2003 pre-NA release old. But in it, it says,

Originally Posted by
SE
Does having a thief increase the possibility of a drop even in key/quest items (aprons, skull, wadi grass, etc.)? Of course! Only the thief has the ability to acquire the Treasure Hunter job ability. There is an increased chance in obtaining dropped items.
The link to this can be found
here.
-I don't exactly know what 'wadi grass', but I do know that the Crab Apron is a r/ex drop.
-As for the "
key/quest items", I must confess that I edited into the otherwiki that TH affects
key item drop rates.
That's how I read it back when. But looking at it now, it probably just means
chest keys.
So if anyone has been depending on TH for VNM abyssite conversions because of this, I apologize

Unless I'm the only one that thinks they're talking about chest keys now.