I know all this. I've tried grouping them when they've been placed on other ship parts. I was using a blank schematic to illustrate the problem. Also if you notice they are not mirrored via the mirroring button. They are two separate parts . If you think I'm incorrect and are willing please do try and see if you can do it and then tell me how. : )
I don't think you understand, in the demonstration video you've shown above 2 mistakes where the first was after creating the first half circle, is that you chose to spawn the second half circle attached to the translucent starter cube instead of as an attachment of the first half circle.
Making it a separate object entirely from your first half circle, remember parts will not connect together from the translucent starter cube. Every part connected to the translucent starter cube is a base part and every base part is a separate object. As any base part connected to the translucent starter cube will not group together or be able to be made a custom part with other bases parts or anything connected to other base parts.
The second mistake you made was when selecting the grouping to make a custom parts is that you selected the parts from left to right. If you followed the steps I provided above and attach a part to the base part it works like a chain where the base part should always be at the end of the chain as all the other links are connected to it in some way. Always select the part from right to left where the very base part of your ship(usually the most left-hand part) should be the last thing selected to have all the parts linked together to the base that will become the center pivot of the part when saved as a custom part.
As for mirroring, I didn't say you used it in your video I was merely giving you a tip.
Perhaps I'll make a video tutorial for tips on how to properly use the ship designer in the future.