I think now I'll need a way to automate some of the process, or it will be a nightmare to do manually.

Here's an example of a mesh weave I did for another chair I modeled:

That one was fully automated. I simply modeled one woven strip, and a maxscript did the rest for me. However, it was easy, because as you can see, it was a planar surface area. This one will be more complex.

Here's the new spline layout for the Cappellini chair, with the green support tubes now running vertically, and the "weave" tubes, in blue, running horizontally. Much, much better, and more sensible design.  Only now, as you can see, there will be far more weaved tubes to deal with:

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