The cabriole leg he did in that video is not typical of what a 4th axis is used for, and there is nothing simple about actually making the indexed toolpaths needed. Several separate files, several different bits used, a mastery of 3D visualization to both create the initial 3D file and then spin it in your mind to figure out how to approach each surface, etc. Ultimately you CAN use Aspire to make all the tool paths shown, but it won't be easy or obvious no matter which program you choose.
Aspire has wrapping gadgets for basic rotary work. It can also import 3D files to either "unwrap" it if not too complicated, or rotate to each face and cut as basic 3-axis 2D and 3D surface work. I could manage all that, but I'm not sure any of my students could. You've got all the basics, and some training. It will demand creativity and an understanding of how to control the 4th axis with your wood block clamped into it to make a similar thing.
Your described project is far simpler. Yes you can both create all the 3D parts within Aspire, and then create the toolpaths needed to make that vex/cave con part. Personally I'd just treat is as two X/Y/Z tool paths, and use a single g-code command to rotate the block 180 degrees before cutting the second side and mortise. G0A180 would probably be all you'd need to type into your controller.
4D