Driving from MNPLS, I know the routes you speak of to get there (avg of 15-16 hrs drive). [I don't feel there is enough extra benefit to drive around and over to the west side]. I put more value in staying some place that I can ride from. Forget driving up to a parking area, or parking along the road. Only time my truck moves is to go get parts when/if needed. There are plenty of great places to ride on the east side, too. Just need to do your home work and find them. And then if you want to "bomb" over to the NW side, you can do that too. The ride will be similar to running from the town of West Yellowstone up to Cabin Creek but on a winding trail.
Most important to understand is that the Snowy's is remote. Meaning you don't have a town of any significance at the base of the mtn (either side). No place to "shop" if too cold, or tired, or broken down while the rest of the group finishes out the day. If you do need parts, they are a 45 min drive away [to Laramie]. On the west, I think you are driving to Saratoga. Not sure if you can buy gas on the east side of the hill anywhere other then Albany Lodge. Just need to plan accordingly.
If that what you like, and set expectations accordingly - it's a great ride.
One more - the Albany Lodge is under new ownership. Former owners were a bit tired of it, and shows. Updates are already under way.