Hi Noseman,
As you suggested, moving away from the particle system, I tried approaching this using a "Cylinder with a Cloth tag" and a "Sweep with a Rope tag". However, I still can't quite achieve the braiding mechanism and the exact setup I need.
If I could somehow create a setup where a "Cylinder + Cloth tag" is continuously generated and extending from an Emitter, I would be able to replicate the mechanism seen in the real-world video and finally achieve the animated scenario I am dreaming of. I hope this sparks your curiosity and we can figure out a solution for this together!
(I also want to kindly remind you of this 6-year-old C4D video: https://www.youtube.com/watch?v=Z4TJp95T6ys. It seems they actually managed to build a continuously growing dynamic rope system out of an Emitter using those specific tools).
If you have the time to take a look, I have shared a Drive folder below containing the "Cylinder - Cloth tag" scene, the "Sweep - Rope tag" scene, and the real-life reference video of the machine.
[https://drive.google.com/drive/folders/1V0rGNqtyClD8PmTZRklo2tmxIGUBSTKc?usp=sharing]
Thanks again for all the help!