I think everyone can agree that one's performance in medical school means more than the school one graduates from. That being said, go to a school where you'll fit in, enjoy the atmosphere, and do well.
I think it's an exaggeration to say that the name makes no difference at all. A person who graduates in the middle-of-the-class at Hopkins will likely do better in the residency match than a person who graduates in the middle of a lesser known state-school. Likewise, I imagine the AOA at Hopkins will also do better than the AOA's at other medical schools. It's not to say that students at lesser known schools can't compete or have a class ceiling, but they do have a harder path to prove themselves.
The career one seeks also makes a big difference. I've heard that for someone who opens a clinic or goes into private practice, the name won't mean much. However, for those that choose academic medicine or want to revolutionize medicine or health care, the prestige and connections that come with "top tier" medical schools will make a difference. Funding comes easier and having alumni ties will help. Again, what one does with those opportunities means more than the opportunities available, but there is a difference.
It's the same story with medical school admissions for most of us now. Some have even heard the line during interviews of "Which school is that again?" I go to a top-25 public university. HMS accepted one student from my class last year, UPenn took 1, and JHU accepted 4 (and it's a public school, so it's a huge pool of applicants). Someone on this forum posted that only one Berkeley student was accepted on average at HMS each year. Nevertheless, I'm fairly confident that at least 10 Harvard ahd Hopkins undergraduates make their way into HMS each year. So you can make it to the best medical schools from anywhere, but you'll need to prove yourself more than some other applicants that have the name prestige and reputation going for them. I imagine much of that mentality carries over to residency matching as well. I don't know if there's a correlation between "good doctors" and "good medical schools," but schools ranked higher often match better on average than lower-tiered medical schools.
In the end, everyone has a different opinion and I completely agree that happiness and medical school ranking probably show no correlation, much less causation. Opportunity is what you make of it and everyone has a chance to cross the finish line. However, I think it's also naive to assume that everyone starts at the same line to begin.