Fuel Engine starts after priming, then dies in 2-3 seconds

For starters take the needle right out of the bottom of the bowl to check you are getting a good fuel flow.
Once that has been established screw it back in till it seats lightly then back it off the number of turns specified in the manual.
Usually that is slightly rich.
However if the engine will not run, back it out a little further , around 1/6th of a turn at a time till it is 2 turns out past what the manual says.
If that is not good enough then strt at the same place and go the other way.

A lot of new parts for old EPA non compliant engines are now compliant so things like needles are longer with a finer thread so the number of turns off lightly seated can be nearly double what the manual said.

My rough as guts method of diagnosis is to spray carb cleaner down the carb throat.
If I can keep the engine running for a minute or so then the carb is elevated to prime suspect.

If not then the exhaust becomes number one as a blocked off exhaust will allow the engine to run a short while till it chokes the engine preventing fresh fuel getting in cause the burned cant get out.
Then there is a gummed up reed valve ( where fitted ) that will allow enough very rich fuel in to maintain running but not enough lean mix to support combustion.

Rivets,

Upon closer inspection, the carb appears to have just one adjustment, which is on the bottom of the bowl. Then, there are adjustment screws for throttle control for both the high- and low-speed settings. That appears to be it; according to the Tecumseh manual, it appears as if this carb is their Series 9 carburetor, which has an idle discharge port/fixed idle system.

First, I got the adjustment on the bottom of the bowl in the ballpark. Then, I adjusted both of the throttle screws, so it ran consistently at both speeds. Then, I went back to the nut on the bottom of the bowl and fine-tuned that. Now it purrs like a kitten at both speed settings.

Thank you so much for your help and input. I'm new to small engine repair and am really glad it wasn't something more complex. Take care~