It is certainly possible to avoid using a cone, and I have also done this in a pinch. Though it works, there is no real benefit to using a bare filter (that is, using a paper filter without a filter-holding cone or other device) as far as the brew/outcome is concerned (other than the obvious, trivial facts of less equipment and not having to clean the cone). There are other reasons to not use a cone, and there are also a few downsides.
The main downside is that paper filters are quite flimsy, especially certain brands and types (e.g., Hario V-cones are very thin as compared to thick, almost fabric-like Chemex). They are meant to be used with a supporting cone or basket. With support, they hold together nicely. Without support, they can easily rip, be otherwise damaged, or collapse when filled with water. The consequences are natural: contamination of brewed coffee with grounds, spilled near-boiling water, or other mess.
One might argue that you have better control over the speed or length of infusion, but the coffee will not drain as well or as quickly. You can precisely control the length of water/grounds contact with reasonable pouring methods. The ridges on the cone filter (like this or that question) help to wick the coffee away from the filter so that it drains better. You can also find cones that have a drain plug (one example), so that you can steep the grounds for some time before draining, but I don't believe in that either. No matter what, I argue there's no benefit to a bare filter for this reason either.
That said, there are reasonable ways to use a bare filter. I sometimes use a bare filter for cold-brewing coffee without other equipment: I use a large basket-style filter (arbitrary example) with some grounds, then tied up with a piece of kitchen string into a small sachet like a tea bag, then submerged in a jar of water (see other cold-brew questions, and my problems with same). However, I still have to be very careful to prevent the filter from tearing when removing it from the brewed coffee.
Your friend is clever and plucky for sure; however, I don't think your friend's method is going to catch on at your local coffee shop as the next fancy brew method.