The gltTF file is generated the same way, regardless of the input is a segmentation or a model tree. Please compare the two files (they are just simple text files that you can open in any text editor or text comparison tool) and let us know what the differences are.
However, it would better if you used a 3D viewer that propertly supports transparent surface rendering (can do depth sorting, depth peeling, or similar methods).
What you describe works but more complex than necessary.
For high-quality vessel segmentation you need high-resolution images. Typical resolution for surgical planning is 1mm. For vessel segmentation (especially for veins) you may need contrast agent.
For your information, Slicer now offers fully automated lung, lobes, airways, and vessel segmentation. There are several active discussions about it see for example here: