Is there such a thing? The usual places show nothing of the kind.

Countersunk-head screws fit flush with the surface of course, but
another aspect of them is that they stay centered in the holes. Is there
a nut with a conical contact area to fit similarly?

I know a lot of nuts have one or both sides sort of a rounded-off, but
the contact area is not nearly the entire possible.... and I know I
could trim some regular high nuts or coupling nuts off if I only needed
a few, but that would be impractical when I needed 50-100-200 or more.
Even less so when I needed smaller sizes--#8, #10....



In a perfect world there would exist 'countersunk' nylon-washer-style
locking nuts.... as long as they're not a mil-spec item where a 1/4-20#
costs a dollar each....