You understand Projmode correct.
0 check intersection by objectdata (real intersection of elements)
1 acad project the object data to current ucs and search for intersection
2 acad project objectdata to current view
Back to issue
Your line has no intersection with the visible spline,
normally you can use EDGEMODE for this.
But my first test failed, i had to change the METHOD of spline (look at spline properties ctrl+1)
Then it works.
I dont know why, perhaps i made a incorrect test, but try to change the method abd if needed edgemode 1 / i am in hurry, isnt well for testing SORRY.
_
Note: your current view is from "behind"
UCS World
PLAN World
Thats the standard config use and view.
Your objects are not at z=0 (nothing to do with issue i think)
EDIT: To correct them try command flatten. You can do that, but flatten destroy splines and recreate them (so good how it is possible) with polylines.