There should be a folder under each of the parts in the assembly that
have the mates for that specific part. This is different than the
MateGroup folder that is at the bottom of the tree. You should also be
able to RMB on each of the parts and select the VIEW MATES. This should
show the mates of that part in a split feature tree.
If I've interpreted you correctly, you want (for example) the 3 mates that
you added when you first inserted the component.
The only way to do this is create a separate mates folder for each component
and move the relevant mates into it.
The system can't do what you are asking, because as far as it is concerned
the "other mates showing up from parts that are stuck to the subject part"
might be the ones that define its position. As far as I'm aware, it doesn't
try to solve mates in the order you applied them, but effectively creates a
(potentially) huge matrix of simultaneous equations that it has to solve in
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