Maybe you didn’t intersect the vertical faces with the bottom of the leg correctly.
This worked for me after recreating the leg.
Move the cutter into place. I made the leg wider than needed to make cleanup easier. No reason to be stingy with the geometry. the waste doesn’t cost anything.
I placed the cutting surfaces in place over the leg component. I moved it down slightly to ensure that the bottom faces would get cut during the Intersection.
Of course it doesn’t work correctly if you leave the cutters outside the component. Why would you do that? You need the cutters inside the component so that they get cut by the leg geometry to leave the faces skinning over the edges of the leg. The faces shown selected in the screen grab below as well as their counterparts on the opposite side are the remains of the cutting faces.
Yep, I’m pretty out of practice and my last sentence had some grammar mistakes. The conclusion is as you said:
1 - ‘Intersect with’ doesn’t create new faces
2 - I was missing some faces but those faces belongs to the cutter object and that’s why I have to work with them at same level (and not outside the component as I was doing)