How to go about creating points at river confluences?

The Intersect tool does not require two inputs. The help for How Intersect (Analysis) works says "Intersect can run with a single input. In this case, instead of discovering intersections between the features from the different feature classes or layers, it will discover the intersections between features within the single input. This can be useful to discover polygon overlap and line intersections (as points or lines)."

You have to be sure to change the output type to Points to get the intersection points created, since otherwise the tool will try to find only overlapping line segments.

"Could creating a geometric network solve the problem? (ArcCatalog > Geodatabase > Feature dataset > Geometric network). Junctions will be created automatically at the intersections of the rivers."

https://geonet.esri.com/thread/21850

Spatial ecology plugin does it simply as well. The tool is isectfeatures (Intersect Features)

http://www.spatialecology.com/gme/isectfeatures.htm

If you specify a single input feature data source, then it calculates intersections between geometries within that dataset.

Or just do as Richard states.