Would agree to start south and work north for all the reasons mentioned - mainly warm vs cold and accessibility in Glacier factor.
Have been to Glacier several times.
East side - which we prefer. We like FHU and stay at the KOA. We looked at Johnson's twice and the only thing it has going for it are views. The roads and sites are tight and the sites are uneven. Others have said the same on here in the past.
Red bus. We are used to mountain driving so have driven the GTTSR side to side many times. The road is good and is safe except for those coming against you who take their half out of the middle.
Here is why we do not take the Red Bus - you are on THEIR time frame and THEIR stopping points. We like to set our own pace and places to stop other than what they do. It is a nice service to be sure; just not for us.
We have been fortunate to have seen many National and other parks - Glacier and
Rocky Mtn are our top two. We prefer the east side as we are hikers and like the low key as compared to the commercial west side.
Enjoy.