Your problem is not the linkage it's the diaphram.
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. 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