You said lace doily and lace table runner. It could be worn but not as it is, it would have to be made into something wearable. A lot of the things you have guessed could be made with it.
Close with doily but not a doily, has something in common with a table runner but not a place mat. Chair back rest cover?
You were on the right track the whole time but you kept naming things that had lace on them or were made of lace but not just lace it's self.
This is the only one I have ready, I just realized I forgot to crop the rest and if I put them up they would go sideways