Having ridden up and back down the west slope 3 times and up and over to St. Mary once, traffic is definitely the issue. Also, you have a much better chance of seeing wildlife early. Finally, if avalanches are still a concern, they are more likely after the sun has had all day to soften the snow.
I always started from Sprague Creek Campground before 6:00 in either mid or late June. The car crowd doesn’t like to hit the road so early. Depending on where you start from, you may have several miles of relatively flat, straight road. People tend to speed there. Saw a guy get pulled over by park police last time I was there. He pulled into the campground and got a ticket or warning.