Ran across this in my testing also. Publisher does not recognize or use fonts that are embedded in a PDF. You have to have every font that is embedded in a PDF open on the computer you are using to be able to safely place a PDF into Publisher. Affinity suggests either rasterizing the entire PDF or to outline all fonts before placing it into Publisher. They also have no time frame on when or if this will ever be fixed. This is a complete deal breaker for a page layout application.
I have been active in the Affinity forums and that is definitely the case. If the fonts are loaded on your computer there is no issue, at least with the fonts. No option at the moment for placing a file and not having it editable at the moment and does not seem to be something they are working on as I believe this is considered a "feature" I had originally thought it was something to do with Office apps, but as you have said it is due to the fact that Publisher cannot see embedded fonts in the PDF.
Still a great product for the price point, but not something I could leave adobe for at the moment.