I had one of these once at an old house. Lived with it for long enough and then in a pique of anger woke up one morning and ripped the damn thing off the wall.
Found that someone had punched the overflow through into the soil pipe and had left a 3 inch piece of pipe protruding, which was collecting toilet paper and every other unspeakable item flowing past.
The soil pipe was, and this is making me wretch as I think about it, solid with erm, soil.
To shift it, I used boiling water, bleach and half a garden cane. Funny how you become immune to these things after an hour or so. My wife came home to find me standing in my underpants, with her kettle covered in crap, the stench of human effluent everywhere, laughing hysterically and pointing at a hole in the wall shouting "It's gone, you see, hahaha, I've fixed it, it's all gone".
I repositioned the overflow, made good and used the toilet trouble free for many years after.
I know this may seem extreme, but there is no inherent reason why a correctly plumbed soil pipe and toilet should block, unless used by some sort of elphantine human being. My tip is to start early, with conviction, as near to naked as possible, rip the thing up, and go for broke. Once all is done, you'll have the best shower you ever thought possible...