You're right. It sort of works if you follow Pandoc's method of simply indenting following paragraphs that are supposed to be included in the footnote. However, (at least in the HTML version of the book) at best they get moved to the end of the book and other footnotes later on the same page are mis-numbered. At worst, all multi-paragraph footnotes simply disappear from the HTML version of the book. How do you get them to at least show up at the end of the book? You need to add an empty page at the end of your book for citations, and then the footnote will incorrectly show up there.
I think what's actually happening is that multi-paragraph footnotes are being treated as Citations. Thus they are only included in the References section of the book, as described here:
For non-LaTeX output, you can add an empty chapter as the last chapter of your book. For example, if your last chapter is the Rmd file 06-references.Rmd , its content can be an inline R expression:
`r if (knitr::is_html_output()) '# References {-}'`
Regardless, if you don't include the empty file for references in your book, the footnote will entirely disappear in the HTML version. As far as I can tell there is no way to get it to show up anywhere if there are following footnotes in the chapter. As such, I'm not marking this as a solution.
I've opened an issue on the Bookdown project.