I use the Polaris oil, it's synthetic and therefore flows in the cold, resists breaking down when hot, and its the recommended oil. That gives me some peace of mind.
Chaincases give problems, less so if you do the maintenance mentioned above post.
In reality the chaincase only contains a link chain, the same as used to drive the cam in a car engine (and the same as used in some 4x4 transfer cases), steel or powdered metal gears, 2 cheap ball bearings, usually sealed and a tensioner.
I believe the main lubrication challenges are the temperature extremes and the friction of the steel chain on the steel or PM gears. ATF is formulated to withstand heat, flows in the cold, but only has just enough lubrication for gears, because you don't want slippage in the clutches. Motor oil is not formulated specifically for the timing chain, but that is one of the considerations.
Any oil is better than no oil, in an emergency I'd use canola oil if that was all that was available.
Your snowmobile is not where you should try and save money. My friend once told me, "Snowmobiling is not for the faint of wallet".
Don't get me wrong, I try to keep my expenditures down, I just repaired my jackshaft by knurling the worn area, modified a part for the suspension to save 80$, utilized some used parts for the chaincase, but I used Polaris oil to fill it up.
I guess what I'm trying to say; is that there are some places where you should not try and save a dime, lubrication is one of them.