I use the Monopod to keep the camera up and convenient, not really for support. Waiting for a bird or animal to move it is a good way to be in position for extended periods. Most people hunt out of their cars but seldom do you take shots out of the windows. It is rude to block traffic. You should pull over, and when you do there is inevitably a little walk to get position. Last year the Rangers were forcing people to move to pull outs and not letting them park on the shoulders. In general the Lamar Valley you will see quit a bit of wildlife and it will offer few quality opportunities. If you get one really good photo opportunity a day you will be doing good.
Yellowstone is just like any other opportunity, you get what you put in to it. If you just car hunt you will get that quality of results and everyone in a vehicle behind you get the same results. Walking and stalking give you a better chance for unique pictures but on family trips it is hard to do.
If you want pictures of Eagles close up and in flight go to the West Yellowstone entrance. The Madison as you enter the park for the first few miles before the mountains will give you the chance to be close. Outside the park just out of West Yellowstone headed South you can turn west just after crossing the Madison, drive in to the National Forest and work the trails next to the lake. Sometimes the Eagles will perch in the trees on the cliffs next to the lake. You can get really close to them but it is hit and miss.
Another bird spot would be the Hayden Valley along the Yellowstone River. You might get to see the Trumpeter Swans.