If you’re thinking of the # of Production Operators and you have a recent [weekly] record of the overtime hours, add the OT hours (OTh) and divide by 36. This will get you close to # Operators needed to reduce overtime hours & operator-burnout. Note, I use 36 because 40 hours x 90% = 36, and most humans are about 90% efficient (in the long term) when working on production activities that require manual & hands-on actions. So when a person is present, they’ll be about 90% efficient, which is not same as utilization. And I round up for any fractional results.
Example: If recent totaled OTh = 140 hours, divided by 36 = 3.89, rounded up = 4 operators needed.
I have found that getting Management “In the Game” will increase the odds in your favor.
To directly answer, “… clearly we are doing something fundamentally wrong. I suspect it has something to do with not being disciplined about following work instructions, and/or not enough oversight by managers and supervisors.” It might be a case of defining who ‘we” is, and the cause & effect of the lack of oversight. I think you’re seeing the effect.
Maybe get a copy of “The Work of Management”, by Jim Lancaster,
[ https://www.amazon.com/Work-Management-Daily-Sustainable-Improvement/dp/1934109029/ref=sr_1_2?ie=UTF8&qid=1520861366&sr=8-2&keywords=work+of+management ], sorry about the ginormous hyperlink. Get your management folks to read it, then get them to DO it. If they care about what is happening in Gemba, they will start showing up there on a predictable basis (read: daily scheduled Mgt Walk About Reviews), asking tough (but fair) questions, and providing the level of help only they can deliver.
Combine with Training Within Industries (TWI) methods, to address things from both sides of the spectrum.  If your management does ‘Management by Magic Wand’ (they wave their hand and The Magic Happens), you’ll forever be frustrated.  If employees don’t have good training (using Standard Work) they’ll find ways that work for them, but not always the best for the company, like the results you’re experiencing now.
Hope this helps a little.