Hi Miles,
I think you have the right analysis and understanding of the problem. Regardless what tools (Intersect, Select By Location, Spatial Join, or Locate Features Along Route) have the option for returning intersections, the underlying computation for intersections should be the same. I am not an expert for LR, but your expectation for LFAR tool seems reasonable to me. When you expect a tool to return intersection points, you certainly can use other tools to verify results. If your pipelines and rivers actually cross or touch within the feature's XY Tolerance (see properties of the feature class), at least the Intersect tool should find all the intersection points; the duplicate points are by design; the Delete Identical tool should take care of them. Could you do the following:
1. Check what the XY Tolerance values are on both inputs (pipelines and rivers).
2. Run Intersect tool, followed by Delete Identical tool. Check if the resulting points are expected. If not, you can share some repro sample data; we can investigate.
Entering 0 for search radius may mean different things in different tools and doesn't always mean 0 is used; some tools use the XY Tolerance internally, other tools may treat it as an option for certain analysis. Generally for lines to be intersecting or touching they need to be within the XY Tolerance to each other. It won't make a difference if you try any value smaller than the XY Tolerance.
The tools with an "intersect" option I mentioned above are simply based on a search radius. This is not in the nature of conflation that I know of. Conflation typically refers to matching two spatially overlapping but shifted datasets from different sources (e.g. State DOT roads vs. local government maintained roads); you wish to spatially align one towards the more accurate one or to transfer attribute from one source to the other. Feature matching - the core of conflation - is not only based on proximity but more critically based on topological structures and similar patterns. As far as I know, the LFAR tool, as it is now, does not operate as a conflation tool. I just want to clarify that.