I can't make the search find census entries which I know exist because I've seen them on other websites. Is it possible to find and attach them to my tree by searching by piece, folio and page number?
No, you can't do that in FamilySearch. I regularly encounter the same problem in not finding E&W census records on this website. You may be aware that the FS records are generally passed across from Find My Past, who appear to have withdrawn their 1871 collection from here. It's not always possible to find records that can be found on Ancestry, or even FreeCEN, on other websites. This is sometimes due to the names being indexed incorrectly / differently during different indexing projects.