Well, I finally have an update on this mower.
I went in and pulled off the new carburetor to see if any of the passages had become clogged, as I failed to clean the fuel tank or replace the fuel line prior to running it before. No issues were found inside the carb, though I think the float level might be a bit high...will have to figure out how to adjust a plastic float (or see if the old metal one would work). The intake gasket is worn, but I saw no evidence that it was allowing air past it. I didn't have thick enough material to make a new one anyway.
I happened to look at the attachment point for the throttle linkage and made a discovery: there are two possible attachment points!
I had attached the throttle linkage to the hole marked with the blue arrow and had failed to notice the hole marked with the red arrow. I then pulled the original carburetor out for comparison.
The original had only the one attachment point! So this might explain why I thought the linkage had been too short. Moving the linkage to the other hole would allow for the extra bit of movement I needed.
So I reinstalled the carburetor with the linkage in that hole closer to the pivot point. With a bit of adjustment to the throttle cable, now the governor arm doesn't run into the bracket at any point.
Before testing this out, I thought it would be prudent to replace the fuel line, filter, and clean out the fuel tank. I am glad I did! The gasket from the fuel cap had rotted out and was in pieces in the tank. There was an inch-long chunk sticking out of the old fuel filter when I pulled it. That might cause an issue! The previous owner had put the fuel filter in line upstream of the fuel shutoff...which I didn't think was quite right. So I reversed the order, making it easier to replace the filter in the future.
I put the battery back in and added new fuel to the tank. The mower fired right up and didn't have the weird revving issue it had before!
I was able to do some adjusting to the carb and was drive it around the yard with no issue.
A while back, I had done some work on the hood, as parts were bent, and it was missing a linkage to make it flip open properly. I reinstalled the hood and it fit better than before.
I also installed a new brake disc (since stopping is important too!). I need to get new brake pads plus a brake plate, as I still don't have great braking ability.
The transmission whines a bit, and barely crawls along in first gear. Not sure why, but at least it is moving around now! I entertained the neighbors buzzing around the yard on the mower. I also tested out the hitch for towing my DR lawn vac trailer...it should work well, but I have to wait till fall to test it out.
I need to order some parts for the mower now that I know the engine is fine: brake pads, brake plate, deck safety switch, air filter, and a few more that I have on a list somewhere.
Now I need to figure out the transmission...being a Peerless, I assume that it used the bentonite grease that was so popular. I wonder if there is a fill plug on it somewhere?