I have only skimmed the comments and plan to come back to read more when I am able. But I did want to pop in and say I empathize with your situation here, as I am another who feels a bit helpless in the face of jewellery (especially necklaces) and frustrated at how others seem to "get it" so effortlessly.
Like you, I had at one time an idea that a necklace might become my "third piece" especially in summer -- but alas, that was not to be and I finally had to acknowledge it. First, I haven't got a clue about how to pick them, and the reason is, I don't truly enjoy wearing most of them! Like you, I'd rip them off the minute I got in the door.
Ironically, the minute I accepted that, I started finding necklaces I like!! I'm not joking. I now have a lovely and distinctive, minimalist choker style necklace that works inside a shirt collar type top. (An anniversary gift from Mr. Suz). I also have a slightly more elaborate necklace with turquoise stones that is adjustable at several shorter lengths. Add a couple of pendants of various sizes on longer and shorter chains -- including my latest "fitness monitor," which is called a Bellabeat Leaf Urban, and can be worn as a necklace, clip, or bracelet. (I mostly wear it as a clip and sometimes inside my clothing so it is invisible -- but it is nice to have an option). Finally, I added big blingy thing full of sparkles for when I need to go out.
All of these are lovely. All are my favoured silver tone metal (some with additional stoens.) and I truly enjoy wearing them. ON OCCASION.
But they cannot be my reliable "third piece." Nor are they my "signature" jewellery. Signature jewellery means earrings (which never make me feel unlike myself), my watch, a bracelet, and sometimes a belt. Also a substantial ring.
Like Gaylene, I love scarves. i wear those pretty easily. But a necklace is another thing entirely and it is not for me.
Why can't your fabulous bags be your "third piece"?