Need help with toilet not draining
What you describe is a "clogged toilet".  Plain and simple.

You say "snake", but have you specifically tried a "closet auger" that's made for toilets?



They are normally 3ft, but if you look, you can find a 6ft model that has an extendible snake.  The snake part of these is much thicker than a standard drain snake and the long, bent handle starts it on the right path.