Most likely just looking at a carburetor problem. If the engine was trashed due to no oil, it would not run on starter fluid or you would hear the rod bearings knocking. This is not to say the engine didn't suffer some damage from the low oil, but it is probably not totally destroyed.

These little engines are pretty tolerant to very low oil/dirty oil conditions. Just a few ounces splashing around will keep them alive. (not happy, but alive)

good luck!

Yes, it sounds like fuel starvation. Since it only runs for a moment on starter fluid I would clean the fuel system from the tank forward. Was the gas in the tank replaced?

Neighbor's Yardman mower with a Tecumseh engine on it, same issue. He admitted he hadn't checked the oil for 3 years. I helped him fix the plastic shroud over the engine and then he asked me to check the oil. The dipstick was dry. I added a lot of oil before it was full. She started right up and ran good. I advised him to not bother changing the oil as it was pretty much all new :).

On the Tecumseh, clean the orifices in the brass bolt holding the fuel bowl on. Pay particular attention to the tiny one near the end that is on a slant. Clean the paper off a wire tie and use it to rod it out. Finish by blasting with carb spray. If you don't use stabilizer and/or other cleaners/additives you can expect to do this about every year.

Thanks for all the tips. The mower runs fine now after I cleaned up the carb again.