James is my brother and he asked me to try to give you a little more insite into this issue. I am actually the owner of a large mower shop for over 35 years.
First it is very difficult to deturmin which picture is the exact one that you need for your linkage without me actually seeing your engine. They made so many different ones. If you could take a digital picture...or 3 or 4... and send them to us, I might be able to tell you a more exact answer.
If I was guessing, I would guess that you have the linkage in the wrong hole....but that is just a good guess.
The linkage has no effect on gas getting into the oil...there is only one reason gas gets into the oil...(Unless someone poured it in there).....and that reason is a faulty carburetor.
There are two reasons that can cause a carburetor to leak gasoline.
1. a float problem :...a float that is set wrong or a float that has a small hole in it letting gas into it....causing it not to float.
2. A needle and seat that is not seating. (Float needle and seat.) The float needle and seat will leak if they are worn or if there is some dirt or gum holding it open.
In any case, the carburetor MUST be completely cleaned and overhauled with a COMPLETE carburetor repair ki. The manual that James sent you will tell you how to set the float and rebuild the carb...as well as information in his his carburetor answer.
The Governor gear could be broken or even just a piece of it may have been broken or even just stuck, but the gas in the oil would not cause this. Sometimes they just break for no reason. Sometimes they break because of overspeeding...like if the linkage is put on wrong. Also if the linkage is on wrong, it can also put excess pressure on the gear and cause it to break. The govenor gear and spool are plastic and therefore subject to breakage..or...(even melting if they get too hot). (Most likely not the case with a snowthrower, but still a cause for failure.)
If the governor arm has ever been taken off of the governor shaft , then there is NO CHANCE that the governor is set correctly. It might be a good idea to take it to a shop
and have them try to look at it and give you another opinion where they can actually see it.
In any case, I think I have given you several things to think about. Remember, to send the photos so we can look at them if you want to have us give another suggestion.