Engine is a Briggs and Stratton 500 series model 10T502-1303-B1. When you start it, it immediately starts running high idle. The throttle is wide open. The throttle can be freely moved by hand, but when it's released it's pulled back wide open. The throttle spring does have one place where the coils are a bent and separated, but I can't tell if that makes it any stiffer than it should be.

I did try some random springs, and it would either do the same thing or it would hunt with the throttle going back and forth between wide open and near closed.

Any way to diagnose the problem?

 

From your description it sounds like the engine was surging and someone decided the cure was to start messing with the spring.The cure is to replace the diaphragm and readjust the spring. Below I will post the instructions for replacing the diaphragm, then in a couple minutes I will post the picture of the spring placement. Have a good one. Geo

You need to install a new diaphram kit, it is Briggs part # 795083 or old # 495770 also Lowe's has a kit #5083 which has the same parts.
Here we go:

Remove air cleaner, there are 2 bolts that hold things together, one on the front of the tank 3/8" and one into the block 1/2", remove these bolts, now "slowly" slide the carb/tank off the intake tube and breather tube, now tilt the tank in to release the throttle linkage and waalaa the carb/tank is removed. Check the intake tube to see if the “O” ring seal and plastic retainer are still on the intake tube, if so remove them and re-install them into the carb. Remove the 5 screws from the carb/tank remove carb(don't loose the spring) now spray all holes, cracks and crevases in both the carb and tank surface with brake parts cleaner, remove the main screen(looks like a thimble), now with a small screwdriver pry out the main jet(carefully) and clean it, check the “O” ring on the main jet for damage, if it is damaged it must be replaced, reinstall the jet, it can be difficult some times to get it to snap back in place(I use the rounded end of a screwdriver handle). Remove and clean the fuel pickup stem(not necessary if the fuel tank was clean). Clean any junk/rust from the tank. Install the diaphram on the tank then the gasket(no goo or sealer) now carefully replace the carb(the spring will try to misalign the diaphram), tighten all screws a little at a time so as not to crimp the diaphram until they are all tight. Install the carb/tank in reverse order and you are done. If I missed something one of the real mechanics will correct me. Have a good one. Geo

 

After you get the diaphragm replaced you will need to adjust the spring. In the pic below the tab that the spring is hooked into is moved forward or backward to adjust the speed. Have a good one. Geo

 

Good advice and pictures from geogrubb. He'll get you fixed up!