It took me a long time, but I finally figured out how to get a thin object to align correctly. ICP alignment is notoriously plagued with dominant plane matching. Meaning if there is a planar surface that has a wider area than the rest of the scan, it's going to try its best to match to that, regardless if you set up point matching. This extends to front and back facing planes and it is very frustrating.
After weeks of trying different ideas, I finally found one that works. I put a white cardboard box on the turntable and put this object (one of my childhood toys) on top of the box. During the scanning progress, I made sure at least some of the box was in view. Without the box, the scan wouldn't align even during the automatic turntable processing phase. But with the box, the scan was perfectly aligned. The render below is the result of only 2 turntable scans of 8 steps each. they aligned with each other really well after I cleaned the box out of the scan.
Render is out of Marmoset Toolbag 5 with a triplanar corroded aluminum material. I didn't capture color data in the scan on account of the spray, so this is the next best thing.