QGIS snapping issue

I can confirm your finding. It seems that even when activated snapping, points do not always snap. When they do and when not seems unpredictable to me. In the screenshot I draw the pink lines first, than the green ones. For the 6 lines on the left side, I started at the bottom, for the 6 lines on the right, I draw them top-down. So from this sample, one could guess that end-point snapping on the pink line is more problematic than starting point, however, there are exceptions.

So it really seems that the lines do not intersect, even though zooming in as much as possible shows the end-point of the green lines always perfectly on the pink ones.

A workaround: create a new layer for the green lines where their ends are a little bit extended to be sure that they cross the pink line. To do this, use the function extend lines from the processing toolbox and set a small distance. If you now use again Line intersections with the extended layer, all intersections show as expected. So this shows that snapping does not guarantee intersection.

Edit: you can use geometry generator to create the same effect, so you don't have to create a new layer. I added a symbol layer on the green layer and defined it by geometry generator:

intersection( 
   extend(
      $geometry,0.1,0.1),
      geometry(
         (get_feature_by_id(
            'pink_layer'
            ,1
          )
       )
     )
)