Need help finding model/year of Snowhound

Hi, everyone! I am at the end of my chain trying to locate the year of manufacturer and real model of my Toro Snowhound. I picked this little guy up for 50 bucks back in 2021 and it works awesome. Just ordered a new belt for it as it is worn. The throttle cable is stuck full open so I intend to replace it wit a new one. It looks to be non original anyway so I want something that is a little less of a hack job.

The seller told me it was a Snowhound 20, Model #31301 Serial # 1800 502033, Tecumseh 4 cycle 3.5 Hp. However, it has a Brigs and Stratton upon inspection and I can't find references to the serial/model combo anywhere on Toro's site. I also use the number at the top of the badge that is 3 078 603 at the top of the badge. He mentioned it was built in 1979 but I don't think they made them that late. The engine code looks to be 1964. Engine model and type is 81202 0170 01

I am unable to share pictures. I teied uploaing but it failed. Sharing a onedrive link failed as well. Thanks for any and all help! Apologies for typos. Did this all on mobile.

Thank you very much.  This is way more useful information than I have been able to locate.  I guess the serial numbers line up with the engine and not the chassis sticker?  That is what was misleading me all along.  I have no idea where the previous owner got the 1979 but when I saw the engine year I figured the build had to be in the 60s or early 70s.
 
As to the throttle cable, I figured I would not find an exact match.  I have been all over parts lists for these things and they are discontinued.  The one on there appears to be not original and fashioned on there to "work" (when it did).  I will try soaking it in motor oil and attaching it to the adjustment side a little better than the previous person did.  Now I do know why the person that sold it to me said he dropped oil in the throttle cable sleeve twice yearly.  I have tons of small engines and never do this to them.  But I will practice it with this one from now on.