My experience has been a little mix of all of the above. I got out in 2011, so my experience has been fairly recent.
Service academies have GREAT networks. I wasn't so sure, until i interviewed with, and worked for a Coast Guard Academy graduate at a PR firm.
So the network DOES help. On top of that, a service academy is a good place to come from. I don't know if that really had anything to do with my major, but SOME people recognize the pressure and requirements of cadets and midshipmen, and that making it through that is a good indication of how you could do in a job.
That said, what you do, or don't do, while you serve also has a major influence. My area of work is LARGELY about experience. Because of my time as a public affairs officer, my deployment to a major oil spill (in a public affairs capacity) and my military public affairs education/experience at the Defense Information School, I had SOME things going for me.
By the time I got out I was also well on my way to my master's degree. It's not NEEDED in public relations or public affairs, but it's a nice thing to have, and it does help with the work I do. I do think graduate school is a nice transition, working with classmates in the private sector; doors can also open with new grad school friends, contacts and professors.
If I had the CGA diploma in my back pocket, that will get a second look for grad school. It might get a glance in an interview too. But without my real world public affairs experience, I'm not all that attractive.
Where am I now? I have a Coast Guard Academy diploma in my back pocket, five years of military leadership experience, advanced training in public affairs from the Defense Information School, a deployment to a major environmental crisis (in a public affairs capacity), almost eight years of public affairs/communications work, and a master's degree in strategic communications. My CGA diploma has helped, but at this point it's much more about experience (including as a military officer, but mostly because my role is related to the work I do now).
We do a disservice to academy graduates when we say that it will be easy, that they will be sought after (no matter their experience), or that their military experience will translate immediately to some lucrative position. Yes, sometimes it will, but sometimes it won't.
While MANY people appreciate academy graduates and military experience, there is also a stigma around it, as well. As a military officer turn civilian, you may be too direct or forceful. You may not play the politics game the same way as it worked in the service.
AND you may be disappointed with the workforce you enter. You get used to the way the military does things, and the relative "accountability" you can expect. It's different in the private sector. I'm not saying either is right or wrong, but they are certainly different.
At the end of the day, your resume is what you make it. There are no free passes when you get out. Think about what that civilian life will look like for you, and if you can do things to build the experiences to successfully enter that civilian life one day, well do them. The military, in my opinion, does a very poor job of actively setting its people up for life after the military. You will have a one week TAPS class, but there is so much learning that could be had on the ride through your career, about that civilian world.