Author Auto-change status from "Complete" to "Reading"
Currently the fastest - realistically the only usable way - to mark all the chapters read is to set the series to "complete." This would be less of an issue if you could simply mark the chapter without opening the link (thus wasting tons of bandwidth on both sides and time on mine). Even better to have a "mark all read" button somehow. But that isn't my real suggestion.

Marking the series as "complete" satisfies this, but it seems there is an issue. If there is a subsequent release, the new chapter also gets marked as read with no further action on the user's part. We may think we've read it, and worse it won't show up in the follows list. So that makes "follows" totally useless in that case. If a new chapter comes out and you haven't read it, then the status shouldn't be "complete" anyhow. It could be automatically changed in this case.

Another related issue. If you mark a series as "complete" to show every chapter as read, then change that to "reading" (in hopes of getting notified for new chapters still), it will clear all the read chapter markers. I have to assume that isn't intentional behavior. It certainly isn't desirable behavior.
Technically the read markers are saved only on Reading and Re-reading, Their visibility on Completed is only a visual effect, the actual markers are not in the database. The reason for this is the massive amount of markers that would otherwise be necessary.

The problems with this system are well known, but until a better solution will be implemented, there's no way around them.
In regards to marking individual chapters as read without opening the link, I've made a Firefox extension that does that (among a few other things). The "mark as read" button uses MangaDex's API call, so it should be very light on bandwidth.

I didn't implement a "mark all chapters as read" function, because doing so would require sending a bunch of these API calls all at once, which might be interpreted as a kind of mini DDOS attack, and even if it weren't, it still would not be kind to MangaDex's servers.

I made a post about the extension here:

If you were at all interested, I could add a feature so that when you finish reading a manga (that is, when you finish reading a chapter marked with an "END" tag), that it would move the chapter into the "Complete" list, but you'd still suffer from the side effects you mentioned about moving manga into completed (that's a server side issue; nothing I can do about it with an extension).