Simulating cloth for instance with spring/damper models is quite old. Even older than robotics research on folding cloths—and it has progressed pretty far. See for a recent survey:

https://arxiv.org/abs/2407.01361

Yes, but you will have to wear a hinged stainless steel shirt that makes you look as ugly as a CyberTruck.

worth it!

Are the Foldimate and Laundroid not meeting your needs?

Can you clarify if Foldimate is still a going concern? Their Wikipedia page says they folded, but some website is trying to sell Foldimates. As for Laundroid, 16000 is more expensive than getting the work done elsewhere.