Can I design or remix a model in SketchUp Make and legally share it via Thingiverse, MyMiniFactory or similar?
Again, I'm not a specialist in intellectual property law (or any other sort of lawyer), but I have a different interpretation.

There is a potential difference between the use of the product, and the use of the derivatives of the product. For example, GCC has a restrictive (open) license, but its OK (in some contexts) to use the compiled code in a commercial product.

Generally, you (as creator of a work) will own all of the rights to a work. The supplier of the tools you use does not usually gain any rights to the resulting work.

So long as you don't gain from your use of the tool, it is probably OK for other people to profit from the use. Sharing with non-commercial should be your safest option if you're worried - but be aware that this doesn't prevent people from taking your design to a print service indirectly.

Their license says:


  Non-commercial use means that you may not sell, rent, or lease the
  output of the Software


This implies that they are not making any claims to restrict how other non-connected persons might use your works. Its specifically talking about use as part of employment, or you making a gain.

I would also note that to me the language in the license does not look to have been drafted with a clear understanding of intent on this issue.