To install the seat, you will likely have to remove the carb. You could try installing the seals and new needle and make it easy on yourself.BTW, I would suggest the OEM Tecumseh parts. It appears the link goes to an aftermarket. Any Tecumseh dealer should have them plus no need to pony up for shipping. Proper order: W/ seat install- you will have to either blow the seat out of the hole via the fuel fitting or dig it out. The new seat goes up into the carb with the ring up. It fits tight. You can use the back end of a drill bit if it is smooth. Hook the needle wire under the tab pointed out as you did before and install the float and pin. Check the float level as shown in the manual. Remember they are doing it with the carb off. Slide the new upper bowl seal on, the lower seal over the cleaned nut and away you go. I'm sure this is in the manual as well for something as a reference.

Assuming that you are dealing with varnish somewhere in the carb, where you can't see it, you might be able to solve the problem by adding gum-out or something similar to the gas tank, getting the engine started using the primer button, and keeping it running by priming it manually every few seconds, when it starts to slow down. I was able to get our lawnmower going this spring, when it would start, but not idle. It takes about an hour of operation for the cleaning solvent to do its work. Eventually, the engine should idle on its own. If you try this method, and you are successful, then you can be sure that it was varnish that caused the problem. There are other possibilities, however. There could be a mechanical problem with the throttle, or there could be an air leak somewhere, causing a lean mixture. There is a slight chance that the carburetor has something blocking fuel flow that will not dissolve, like a metal filing, or some other foreign material.

If your float bowl has a "pinhole" leak, then the bowl must be made of formed steel, plated with zinc or something to retard corrosion. It might have rusted from the outside in, or from the inside out. Maybe you can tell by inspecting the part under a good light. It seems like the Seafoam is working for you. Other people have had similar success with a variety of different fuel system additives, "gum-out" and the like. There is a possibility that continual use of these additives will cause polymer seals and o-rings to degrade over time. You can reduce the risk of this happening if you use minimal, and recommended, amount of additive, mixed with the gas. This repair took extra time and effort-that's just part of the game. Every mechanic can recall stories of repairs that seemed to take forever to accomplish.

"Small Gas Engines: Fundamentals, Service, Troubleshooting, Repair, Applications" gets highly recommended by two reviewers at Amazon. The link below has a couple of other possibilities too. It's $57 (used as a college text) but seems to be written for the beginner. Amazon offers a package with the Briggs book for $67.85. Reviews on the Briggs book complain about dated information and lack of diagnostic and overhaul information. If you can find the right book, it sounds like a great Father's Day gift. Good luck. Here is a link that might be useful: small engine books