I don't think you will find a single word, but there is a good phrase, go for the jugular, defined by Collins as:
to make a savage and destructive attack on an enemy's weakest point.
Examples: You smelt blood and went for the jugular. Times, Sunday
He spurned the trivial and went for the jugular. Christianity Today
Bath spurned the three points and went for the jugular. Times, Sunday
(The comment of @Tom22 just registered on my brain, but comments don't take precedence over answers. More importantly the OP may not want a phrase.)
Single word for the bison example or the jousting example: There are two parts to bringing down the bison with arrows. First is to know where the vulnerable spot is; second is to be able to hit the vulnerable spot with enough force to bring the bison down.
The first is calls on ancient knowledge or tribal knowledge or some such phrase. The second is skill born of long practice.
These two factors combine into expertise. Merriam-Webster:
special skill or knowledge : the skill or knowledge an expert has
The comparison with bullets should be sharpened. A bullet aimed with skill at the right spot will also bring the bison down. A bullet aimed badly or with lack of knowledge of the right spot may just wound the bison.
I suggest your sentence be rewritten as:
In a demonstration of expertise, the native Americans could fell a
bison with their arrows by hitting the critical spot, although
bullets, badly aimed, would fail to bring the beast down.
This fits your bison example, and it would also fit the example of jousting. For your example of
ability to find a weak spot in someone or a system
another word, such as perception, is needed, and two answers have already supplied it: the answers of @Shane and @Stephen Santa Fe.
In summary, I doubt very much there is a single word fitting both situations -- hunting or combat, and social competition.