Do you think this might be possible with Zotero after all?
Use Hypothesis to highlight/annotate. Optionally import them into Zotero with the Hypothesis-to-Zotero tool, if that's where you want them.
Do you know if you can do full text search for saved webpages? That would be useful.
Yes, Zotero can do full-text search of PDFs, webpages, and text files. It can also sync the full-text index separately so that you can do full-text searches without keeping the files on every device.