Does branch of service really matter for skills translation into civilian jobs?

No, I don't believe that you branch of service matters much. And that's especially true with increasing joint duty

I think job role only really applies when you're applying to a field you worked in on the military side. In those circumstances, civilian certification is the gold standard. Sure, the Corpsman may have assisted in the clinic with hundreds of procedures and the medic has done a thousand IVs and minor surgeries. But, on the civilian side all that is irrelevant (and illegal) without the correct certification.
For the other non - military oriented job, I think leadership, education, and resourcefulness are the main employability characteristics we have across all services. The higher rank and longer the service, the more it looks alike from one branch to another. An E7 or E8 are remarkably interchangeable between branches and in joint assignments. While we can tell the difference, to the civilian sector it looks the same across all the branches. Frankly, it looks like competence.

Depends on many things. I'm going to touch the "Third Rail" here and offer my insights...such as they are. Let's start by excluding all of the higher-profile "contracting" jobs, because obviously, if you're hired on to manage an airframe maintenance program for a foreign ally...it makes a difference whether your military experience was on that airframe or not. The same could be said for going to work for a defense contractor optimizing or developing the next generation of a platform you have years of experience with. In the private security industry, I would assume that a "Tier One" Delta operator or DEVGRU veteran are differentiated primarily by their contacts within the "Industry", and who makes the decisions on hiring. For the rest of us, I don't think most Fortune 500 companies can differentiate between a former Army captain and a former Navy lieutenant, provided both served in generalist roles. If a Marine Corps E-6 is applying for a management job alongside and E-6 from the Air Force, it "may" make a difference if the Marine led groups of junior people, and the Airman worked as a highly skilled member of a team of relatively senior people. The more likely scenario is that it's going to come down to evidence in either members' record of technical training or certification applicable to the job being sought. The key take-away is that the civilian sector (and by that, I mean primarily the private sector) doesn't GENERALLY make hiring decisions based on any unofficial "pecking order" we recognize in the Military. There are naturally exceptions; the Park Service, for example...seems to prefer hiring Soldiers or Marines for their field maintenance crews (a "feeder" for operations positions) as to Airmen or Sailors. This makes "sense" as a general rule since the job involves long periods of work in the outdoors, possibly from a base camp. That being said, they tend to hire Park Rangers based on the prerequisite of having a bachelor's degree level of education in natural sciences. In that sense, a twenty-one year old NYU grad with a degree in botany probably has a better shot at it than a Navy SEAL with a B.A. in political science.