Kathril, Aspect Warper and his ability's "targets"
Your general understanding is correct; this ability does not target, because it doesn't use the word &quot;target&quot;. It IS still a normal triggered ability though.

603.1. Triggered abilities have a trigger condition and an effect. They are written as “[When/Whenever/At] [trigger condition or event], [effect]. [Instructions (if any).]”

Your confusion seems to be that you were under the impression that all abilities (or at least all triggered abilities) target. This isn't the case. What we have here is simply a triggered ability that has no targets.
Because the affected creatures are not targeted, you do not have to choose them when you put the ability on the stack. Instead, you choose them as part of resolving the ability.

608.2d If an effect of a spell or ability offers any choices other than choices already made as part of casting the spell, activating the ability, or otherwise putting the spell or ability on the stack, the player announces these while applying the effect.

So when Kathril, Aspect Warper enters the battlefield, its ability triggers. The next time a player would receive priority, that ability goes on the stack. Because there are no targets (and it is not a modal ability), no choices are made as part of putting it on the stack. Players will receive priority like normal and could respond to that ability. Something like Willbender cannot be used here, because the ability doesn't have a target, making the ability an invalid target for something like Willbender.
When both players pass priority, the ability will resolve, at which point you follow the instructions, making choices as needed.