Bad experiences with the production of an accepted manuscript
It doesn't matter whether you paid an APC or whether the publisher is getting money from subscribers for seeing your article.
There are journals that have very significant production backlogs, so that accepted papers are not produced for months or even years.  A long production backlog is not really cause for serious complaint, though undoubtedly both you and the editors of the journal are not happy about it.  Note that frequently the only way to fix a production backlog is to stop accepting papers (or accept very few papers) for some period of time.  (In particular, back when people mostly read papers by going to their physical library, there was no reason for the production backlog to be any better than the publication backlog, and some publishers still think this way.)
However, the number of errors in the production process and the amount of time and number of iterations it has so far required to fix them is completely unheard of in my experience.  I have had some production issues twice in 20+ papers (both times with the same publisher but different journals), but they were nowhere near what you are describing.
At this point, you should let the editors know what has happened and ask them to complain strongly on your behalf.  (Of course, you should be nice and understanding about it, since it's not their fault in any way.  But they have leverage that you don't.)  They are on your side.  They can see you have been treated badly, and they know that incidents like this are very bad for their journal.  This should be a serious issue, to the extent that, if these kinds of production issues are a common occurrence, the editorial board might try to switch publishers (if they own the name of the journal) or resign.  (You don't know if your case is a one-off or common, so you can't very well ask them to do this, but they should want to know and should take this seriously.)
Finally, I think you are well within your rights to not submit to a journal run by this publisher until you have some indication they have improved their processes, and you should quietly let anyone who asks you for journal advice know about what you have gone through.  (If someone reads this question, figures out who you are, and e-mails you, you should let them know who this publisher is.)