This is a genuine question - I think you know where I stand... where the hell are you getting your "thinking" from? I'm guessing your absence records operate similar to ours - a system common in the public sector. The first set of information contains the dates, followed by the number of sick days. Then there is a drop down list of (a remarkably stupid long list) of conditions (that scrolling through makes you lose the still to live). There is no "sickness" / "accident/injury" distinction, and there's definitely no "it's our fault, don't say anything" category.
And being brutal, "work (related) stress" may be a real thing, but no doctor is capable of diagnosing it. The doctor has your word for what is happening at work, and has accepted work is the problem. They have no proof of that. They have proof of the symptoms (perhaps) but not the cause.
I'm going to give you an example. It's the truth. It's my truth. Three years ago I got a new manager. Long story short, she wanted me gone, and I knew that would happen from day one - there was "history". She spent 9 months trying to force me out - I am tougher than that. With nasty disabilist bullying. When she couldn't she tried to make me redundant even though I wasn't. I contracted immediate work related stress. My GP signed me off for four months whilst my union obliterated management. I was lying. I was fine. Faking the symptoms is easy. I miraculously recovered when management surrendered and I was moved to another manager...
A year ago, one of my staff contracted work related stress due to the fact that I was bullying her. Her doctor said that. Unfortunately both I, and IT, had proof of the gross misconduct.
I'm not for one minute saying that you faked anything. And actually what was done to me might have pushed me over the edge - it's not a "tougher than" competition . I'm simply illustrating that truth depends on perspective. Your GP only has your word, as did mine...
I really want you to be able to move forward here. Sometimes "forward" isn't what you think. Your employer is making clear what their version of "forward" is. You will hopefully beat them on the claim. But there no version of forward that includes your continued employment here. I wish there was because I can see how important it is to you. But it isn't going to happen.