"So it sounds like there's no way to Simply select the entity (body) on the
screen directly, then use the shortcts (like in every other CAD program)?"
Not true, even with the outrage added.
There is an important concept to understand here: There is a hierarchy of objects in Fusion, and you need to understand what you are selecting for different operations. This is the list of the things that are selectable in Fusion:

Each of these selection filters can be turned on or off at any time. The other thing to realize is that a Body is composed of Faces, Edges, and Vertices. When you say you are selecting a body in the graphics area, unless you are doing something very specific, you are just selecting its faces, which are not the same objects. Some operations (including copy/paste) are not available on faces. An easy way to select a body is through the browser, as @guenther.andresen says. But, that is not the only way to select a body. You can absolutely select a body in the graphics area ("on the screen directly"). You can do it by disabling all but the body selection filter, or by using Select Other (press an hold the LMB), and going to the parents tab and finding the parent body. See the screencast below
Jeff Strater
Engineering Director