Sorry for my radio silence. I read your replies and was thinking and testing things so I could give a proper response. I’ve tested quite a few apps over the last couple of days so just jotting these notes down for anyone stumbling on the thread:
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highlights.app - beautiful interface, doesn’t remember your reading place (when using an iCloud folder, I dont know if the behaviour is different depending on source). This makes it a non-starter for me, I’m not scrolling through big reports or books each time I close them to do something else.
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GoodReader - wild card entry as I used to use this. Asks to confirm each time you make a highlight, which is ridiculous, so nope to that!
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PDF Viewer - Seems nice enough, but I couldn’t see how to set up local file storage? I set it to a folder in iCloud. (More on this in a minute.)
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PDFExpert - has a lot of menu options I don’t need, but you can remove them from view which is very cool. Has in-app storage as well as storing in iCloud, etc. Seems like all the things I want to do are in the free version. The sharesheet also seemed more rigorous than other apps.
I haven’t settled on a solution yet. @DrJJWMac made a similar comment as I’d made that really this isn’t a tool issue, it’s a workflow issue.
I did wonder about setting up a separate database just for reading, but for whatever reason I’m not keen (I don’t really know why, but I don’t seem to like having lots of databases so it’s probably that). I’m holding this as a backup idea I might return to though.
So for now what I’m gradually doing is stopping use of the “mark unread” function in DT/DTTG. I know I was one of the ones who asked for it 
Instead I’ve implemented the same trio of tags I use in my booklist app: high priority, medium priority, low priority (I already had “read next” and “currently working on” tags that I will continue to use). Tagging like this means I don’t see all the unread files when I’m navigating about, but also means there’s some sort of triage already taking place in my reading list. (I will probably never read the low priority stuff, but at least it’s there if I get the chance, and I can see already that some low priority stuff would probably become high priority if the situation changed.)
I have also sent some files straight to PDF Expert to read directly. I’m not sure I like this as a solution after all (putting unread PDFs in another app/iCloud), but it is a nice way to be intentional about a few PDFs I want to read next so we’ll see.
I still have a problem of PDF books and how to store/read them, I’ve not figured out that bit yet. I don’t really like them mixing with reports and papers - they’re much longer reads and deserve to be stored with other books, but since Kindle isn’t nice about PDFs and Books is terrible for this, I’ve never satisfactorily solved this problem.