Well, I guess you going to have to start at the tank. remove the hose at the carb. and make sure gas is free flowing down hill and the hose is not dry rotted. Next your going to have to take the carb bowl off (if you see bunch of junk (GUM) in the bowl). you next step would be to Clean the main jet and the bowl out. IF that don't work you will have to remove the carb clean the passages ensure the needle will unseat and allow flow through the carb to the engine.

If that don't work you will have to clean the carb. passages using carb. cleaner, compressed air and some times poking wire through offices Fuel or air should flow through the carb.

I usually connect hose and blow on the hose (with my mouth not using high pressure air) if no pressure builds up while blowing on the hose you know the needle is unseating then I flip if up side down and the needle should seat and no air will get to through the carb. which means the float is working.

assemble the carb. being careful with putting the bowl on due to the small 0-ring easily pinched that seals the bowl to the carb housing if you pinch it will cause massive fuel leak and don't over tighten the bottom bolt looking main jet assy. if so you will crush the bowl.

put gas in the tank may 1/3 full. look for gas leaking around the carb. bowl. If no gas is leaking attempt to start it. Hopefully fuel is now flowing to and through the carb. and the engine should start if it don't have other problems.

Looks like its a commercial with the Suzuki 2-cycle engine. Do not use starter fluid in it - it has no oil, and every time you do that you are possibly damaging the engine. Get some kind of spray bottle and shoot fuel/oil mix into the air intake to run it. You need to oil the cylinder immediately - left oil-free, humidity will cause the iron rings to freeze to the iron cylinder sleeve. Either run it on the mix as above or remove the spark plug, tilt the front up and put a little 2-cycle oil in. Leave the plug out and spin it a few times to distribute the oil, then replace the plug. It'll smoke next time you run it for a few minutes.
The Toro site definitely shows a primer bulb on the front of the air filter housing. You will need to find where it goes and determine if it needs to be replaced, and see if the tubes associated with it are attached and in good condition. No wonder he wore out the starter cord trying to start it without a good primer bulb in it. Without the proper parts, the engine will just suck air.

Your mower is a consumer model with the Suzuki engine. Same engine is used on the commercial (Proline) models. Main difference is in the wheels (steel w/ ball bearings), transmission (heavy duty), and handles (again heavier). Forget what else. Got rid of my Proline's as parts became every more expensive, and the mowers were getting to be higher mileage.

Good news is that Suzuki engine is a great motor. Bad news is that parts have gone from expensive to breathtaking.

Sounds like you have a carburetor problem since you can light it up with ether spray. However the Mikuni carb can be difficult to clean properly as passages are hidden. Also, although you didn't mention, if you have float level problems or leaks, replacing needle valve and seat can be expensive and difficult.

Replacement carb is about $250 new and generally go $75 to $100 on Ebay. Replacement coil (when it fails) is $185 new and $75 or more Ebay.

Suggest you tear down carb completely and give it a good soak in commercial quality carb cleaner (the gallon can with dip basket). The spray stuff is next to useless. Try to clean out any passages with a fine wire or welding tip cleaners. Understand there are some passages behind welch plugs that you cannot get at, and will have to rely on the soak.

If successful, upon reassembly you should be good to go. If not, the only solution will be a replacement carb, either new (ouch) or second hand.