I am going to qualify this with the fact that the common ground I have found with my wife for outdoor activities as a trip is camping in a trailer accompanied by biking and hiking. It's not hardcore backpacking or getting entirely away from civilization. However, we really enjoy it and I am thankful for that time together.
My wife and I went up last summer in our camper for our 30th -- not birthdays. We had an awesome time. We went the beginning of June, so couldn't drive straight over from the West side to the East side. Yes, there were reservations to get in on the West side (not the East), but if you camp in the park, that doesn't apply. You can complain about the reservations or you can make one and enjoy the reduced traffic. In general, if you get out early mornings or in the evenings, you'll avoid much of the riff raff.
Found a four wheel drive road in the northwest part that got us to a remote lake and hike. There is a cool little town before you really hit the rough stuff where you can get a bite and beer. For that day, we saw very few people, but did meet an overlander type from Denmark with a very cool Hi-lux based rig at the trailhead and then later ran into him at that bar where we showed him some American hospitality and bought him a beer while listening to some of his travel tales. We took some huckleberry pie to go. That is never a bad call.
Since Going to the Sun Road was closed, hiking traffic where the road stopped was heavy, so we skipped it. However, we found plenty of other hikes to do. We (mainly me) were a bit bummed out at the lack of wildlife on our side other than the mule deer wandering through our campsite and ground squirrels everywhere. We did get reservations on a lake cruise and that was cool, if not our normal thing.
Just outside our campground was a gravel road that we biked on. That same day we biked it, we went back in the truck in the evening with wildlife viewing in mind and saw our first wild grizzly. It was an epic twenty minutes viewing a young grizzly thirty yards away demolishing a log and digging deep for just one ground squirrel. Knowing we'd biked past that spot hours before added to the experience a bit. They hadn't yet closed that road due to bear activity. Oops.
We only made it to the east side once as it was a haul with the park road closed. However, we had a black bear cross just in front of us shortly after entering, so that was also cool. Mountain goats at the lick on the way added a bit more to that leg of the trip. Very cool drive over there if you like driving and seeing country. There are also some good hikes off that road.
Based on our limited experience, the eastern side is better for wildlife and offers access to the adjoining Canadian park to the north, but is definitely more remote. The west side offers more comfort stuff if you want dining and the like -- we didn't since we were in a camper. It also offers access to some pretty remote driving options for a national park that I haven't run into elsewhere.
We did much more, and I won't bore you further with details. However, I've been to many national parks including Yellowstone, Grand Teton, Rocky Mountain, most of the stuff in Utah and many more. Glacier was worth it and we will probably go back. Have done Rushmore a couple of times, and that is a couple of times I wouldn't do again other than having my kids check a box, although Custer State Park is pretty cool and the Black Hills are beautiful in a smaller way than I'm used to living in the Rockies.
For what it's worth, I drove almost 3,000 miles almost entirely off interstate highway on this trip from Grand Junction, CO through the Royal Gorge up to Bozeman to meet a friend and go bear hunting with him, back over to Billings to pick up my wife, then to Glacier and then went home through Idaho to go bear hunting with another friend near Ketchum. I think I can say I saw some country and still found Glacier to be pretty impressive.
Nothing is perfect, but if you want to just avoid stuff because other people are there, give up now. The National Park reservations system is here to stay, so take advantage of it rather than fear it. When it comes to the Black Hills and Rushmore, you will need to like being with others... always. As everyone says (or should), YMMV.
