If it will run with starter fluid sprayed directly into it, then it is still your carb at fault. You need to disassemble the carb from the mower and take it apart and spray everything down with carb cleaner and get it good and clean.

Those older Toro's with Suzuki two cycles are great little mills but unfortunately they are fitted with finicky Mikuni carbs that have to be ultra clean to run. It sounds like there is still some gunk in the main jet... disassemble and try again. You may end up doing it multiple times.
Good luck

Well, you both were right.

Shortly after I had posted, my stubborn self went back at it. Cleaned it AGAIN, re-assembled, and squirted some more cleaner for it to start, and ...she kept on a running!!! Rough at first, yes, but she was running.

Cycled it through about three warm ups/cool downs, and with each time it was running smoother. The fourth time, I put the bagger on and went to cutting grass.

I just can't get over the fact that this baby is 26 years old!! It cuts smooth and even (like it should) and purrs like a kitten. Want a find!!

Yup , I had one given to me , I used it for a few summers until it got tempermental. Then I got the IH sitdown going and never looked back. It languishes behind my shop. My backup mower is the same but for a 'Zuki 4 stroke. I have another one with a Tecumseh motor. Plenty o parts for a spare mower, All for free. :)