@rioville When you select “vertices only” you are generating something like below:

As captioned, there’s no quad (4 sided face) available so that means there’s nothing to use the loop-cut tool on, as it only relies on quad.
To fix this, you could for exemple select vertex A and B and join them (J key), do the same with vertices C and D and then you have quand that are “loopcut-able”. Be aware that you will still have triangles on the corners:

You have similar issue with the cone also because it’s only trias based.
Hope that helps!
Cheers,
/Thibaut