Missed this in your answer. I was just getting to that with my own post.
This is because of where the boot record will be. Booting with the original drive in the system along with the clone, means the clone is not recognized or there are issues. As mentioned above. You will have to clone again. However, again, as others have covered, if you're going to gift the PC, then doing a fresh install from scratch is the preferred method.
Sorry if I'm repeating myself at this point, but most of what I've been doing with the old and new SSD invovles hooking them both up to a second PC that boots from it's own HD and then I just work on the old and new SSDs with an imager or Disk Manager. Also, as I tried to explain in my OP, I've also booted into the old PC with only the new SSD installed along with a USB thumb drive. After using True Image to boot from I've tried using that software to restore images of the partitions to the new SSD.
As for a fresh install, the issue for me is that I bought Win 8.1 over ten years ago and installed from that but then upgraded to Windows 10. So for me a fresh install would mean installing with Win 8, using my key, then upgrading to 10.
Least someone accuse me of piracy again, it was at one time, free to upgrade from Win 8 to Win 10. I don't know if they give you a new key or if you Win 8 key got translated for 10. Then there's the problem with the partition structiure, size, and order which has not been the same with Window 8, 10, and 11. Also, the PC is over 10 years old so a fresh install would me downloading all the proper drivers for the mobo, vid card, etc. Then there's all the tweaking. It's a much bigger deal to fresh install than just restore and image.
Also, I don't follow your logic of being concerned about data on the PC, when you are gifting it to a family member. Are they experts in data recovery? Are they going to hack you or something? I guess the answer would be no. Maybe those concerns are valid for you, but it seems like a little much.
I tend to overbuild or overworry when it comes to things I don't understand well. Just buying a new SSD is a cheap solution for peace of mind and I can still reformat the old SSU and use it on my new one, so I'm not really out anything.
It may not be common to ask what I'm asking for, but I have to believe it's common for people to suddenly realize the have a failing SSD and want to transfer it over as fast as possible.
Anyway, I'll come back with some images.