I'm thinking you could use depot planets and lots of freighters / mass drivers to get around this.  Assume all mineral inputs are routed to earth. Give it stockpile levels, so all spare resources are shunted to Luna via mass driver.   So Luna represents excess minerals.  You want to do sorium refining at Mars so you have a freighter running Sorium to Mars constantly.  Mars also has stockpile levels set for Sorium, and shunts its excess to Phobos. Another freighter runs minerals from Phobos to Earth, so Mars excess is returned back into the distribution network at the top.

In my last game, I had 4 massive industrial worlds (1+ billion) after a couple conquests and extensive expansion.   The main annoyance I had was that if I wanted to install a PDC, for instance, first I needed to remember to pick up the PDC and deliver it, but also route a piddingly small number of minerals there.  The same thing happened with maitenence worlds-- unless the planet had 0.1 accessibility on every mineral, you must make supply runs or the maitenance world shuts down every couple years or so (unless you stock it with like tens of thousands of units, minerals that would be better used industrially).  Lastly, lets say I started a new major construction project-- one of the planets might crash on Gallicite, so I have to rush deliver a bunch of units in short order.  These things happened with increasing frequency at later stages, resulting in that this is all I ever did.

My current network setup is to dump every single mineral on earth and set reserve levels everywhere.  A freighter fleet comes to earth and picks up say, 100,000 of everything.  It then flies from planet to planet, dropping all minerals, then picking up all minerals-- picking up only to the reserve levels set at the world, then flying to the next world.  In this way, to alter the abundance of minerals on many various planets, I need only change the reserve levels, and the system will sort it out after a few cycles.  If I want to put up a financial world, i need only use civ industry to drop the factories, and set new reserve levels for the needed minerals.

Occasionally the network crashes-- its unable to drop of or pick up any minerals at its next stop, so the cycle crashes and the order list halts.  Then you have to manually fix whatever the problem was. fly the fleet back to earth, then put the long list of orders back in to get your cycle running again.  You can use short distribution runs, but that sort of defeats the purpose of the exercise in many ways.

[...] would it be more effective to seperate the responsibilities of an empire among multiple planets?

From my experience lack of population is the main factor for decentralization. What You propose is IMHO too much time-consuming in long run even if it would give some economic advantage.
Placing research facilities and mining installations off-central world is normal, but almost all other is counterproductive because You spend too much time to micromanage individual worlds.

My colonies start specializing rather early.  Since I industrialize earth heavily, and invest in huge construction factory and shipyard production, other aspects need to be offloaded quite rapidly.  In my last game, I started developing research worlds, which took a huge pressure off Earth's industry.  I had one on Mars, and two on alpha centauri.

Because of my emphasis on what boil down to essentially Forge worlds, I need high pop centers to support massive shipbuilding and industry.  It takes a very long time to get secondary worlds up to 500M+ populations, but thats a big milestone and a big step towards getting them producing.

I've been working towards a material distribution network for some time, and have begged steve a couple times for some enhancements to the Reserve Level functions such that I can set up freighters that cruise around and refill planets with minerals to the reserve level, using commands that won't get stuck (like when the fleet has none of X mineral) and require me to redo the whole network every time the cycle crashes.  The idea being that I can centralize all minerals to come to a certain world, then distribute them outwards from there.  Since I do want Sol to be my primary shipbuilder world, this center is usually earth, and then the distro runs go outwards from there.  I see no real advantage (other than role playing) to doing ore distribution or "processing" anywhere other than Earth (or other centralized planet).

Another big one is a Refining world.  I want similar network distribution commands to enable me to fill every planet to a certain fuel level with my tankers, which cycle around the empire stopping off at all the worlds to keep them topped up as necessary.  However, I try to centralize the majority of my refining capacity.  

Maitenence worlds I also tend to do at minor worlds periodically.  Once the ships get greater than 50k tons, the maint facilities start using too much pop at the major production centers, so i offshoot them to smaller colonies focused on maitenence and (for the time being, until I get those fuel network commands (someday!)) refining capacity.  I am not keen on orbital maintenance facilities, never felt I got enough bang for the buck, but once commercial shipyards (and orbital habs?) get large enough you can actually run a lot of pop-free maintenance.

For some of your other points, like administrative and military training, those don't really eat population, so I see no reason not to put them anywhere you need them/want them.