The Gemara concludes that Sandal HaMesumar is not meant to be worn even in one’s home on Shabbos. Why is this halacha discussed in our Mishna amongst the halachos of carrying?
Why was Rebbi Chiya hesitant to publicize his halachic opinion, out of concern for public reaction?
The Mishna, and in fact the Perek, is discussing what you may go out with. In general, things are remembered better when attached to outward similarities rather than the underlying reasons.
For example, the Halachos of Erev Shabbos are grouped together, whether it be a Gezeira that we not put on certain things that might be carried on Shabbos, or that we not eat after Mincha time.