Mowing a badly inclining lawn
Several options come to mind:


Mow across the slope instead of directly up and down the hill, assuming it's not so steep that the lawn mower would tip over.
Use a weed eater to trim the grass on the slope.
Regrade the yard to reduce the slope.
Replace the slope with a retaining wall.
Replace the grass with a mulched bed.


I suspect if the slope is so steep that a self propelled mower would tip over, then a more expensive mower will just give you something more expensive to break.