True, but it would possibly do so in the form of odd artifacts, unless it was shaped like an actual reflector... but then it would no longer be a "pure flood."
If it was a perfect cone, it would pretty much just centre-weight the beam.
Personally I make my own cones from cutting/rolling/sticking rings of silvered plastic film together, and though they're not totally artifact-free due to distortions in the cone, imperfections are only really noticeable on a white wall test. Even with small reflectors (final size ~12mm diameter, ~3mm deep, I get a ~2-3x boost in general forward intensity over a decent angle with no obvious 'edge' to the boost in real-life use.
I'm sure a commercial cone could be pretty much artifact free, if that was thought to be necessary in a practical light, and/or slightly matted to make the effect less centrally focused, if desired.
I have to say that an augmented flood seems generally better in use than a 'pure' or 'naked' flood.
The worst likely practical situation is reading, and just checking now, all that happens there is a slight variation across a page which I wouldn't notice unless I was trying to, (and which I hadn't previously noticed) and which wouldn't annoy me unless I decided to make it annoy me.
For actually doing work, it's at least as good as a pure flood, and decidedly better for any kind of movement, lighting the ground up more evenly as the beam centre naturally points at relatively distant objects.