I think bert and slomo have it summed up rrally, check valves/pushrods and make sure it's actually getting fuel to the engine and not just to the carb.
It certainly sounds like it need fuel in the engine if it runs on starting fluid, which would also point towards the valves working correctly (but bear in mind that theres no guarantee they're working well enough to run on straight fuel just from this).
If it was me, I'd start by checking fuel is getting to the carb, then check valve clearances (to make sure they're opening enough to allow enough fuel in to the engine to run).
If this all checks out, you need to check if fuel is getting to the engine (which sounds like the problem). This could be as simple as a carb clean/rebuild, but you'd need to make sure it's nothing above before wasting your time rebuilding the carb for no reason )especially if the mower isn't very old).
It certainly sounds like it need fuel in the engine if it runs on starting fluid, which would also point towards the valves working correctly (but bear in mind that theres no guarantee they're working well enough to run on straight fuel just from this).
If it was me, I'd start by checking fuel is getting to the carb, then check valve clearances (to make sure they're opening enough to allow enough fuel in to the engine to run).
If this all checks out, you need to check if fuel is getting to the engine (which sounds like the problem). This could be as simple as a carb clean/rebuild, but you'd need to make sure it's nothing above before wasting your time rebuilding the carb for no reason )especially if the mower isn't very old).