Correct, here's the quote, pg. 1:
On the frontiers, extensive home rule provisions allow planetary populations to choose their own forms of government, raise and maintain armed forces for local security, pass and enforce laws governing local conduct, and regulate (within limits) commerce.
It states "extensive home rule provisions", not independence, autonomy, or non-interference.
As far as I can tell, the original vision was:
Planets have a local government that they can essentially organize themselves within limits. For example, it is as if Dresden could determine its own way to determine its Lord Mayor, while he Subsector Dukes were a regional government equivalent like states, like the Governor of Saxony. And the Imperium is the federal government of Germany. But we don't actually know the laws that delineate who is responsible for what.
But apparently it is difficult to accept that "feudalism" is a misnomer in talking about Charted Space, so authors have increasingly had difficulty accepting that titles other than Duke, Archduke, and Emperor are honors, not jobs. The original descriptions of Viscounts, Counts, etc was that their fiefs were scattered over multiple worlds, which was a common historical practice to (generally unsuccessfully) keep lords from consolidating local power. Was that the intent with Charted Space? It doesn't say.
Personally, I doubt that the Imperial government is appointing the planetary governments even in the Core. But I expect that if you have a wealthy, well connected family situated on a planet, they are going to do what wealthy well connected people generally do, which is leverage that into more and more local influence. So you end up with situations like the Marquis of Aramis. The fact that they are the most important oligarch on Aramis is not a du jure result of them being Marquis, but it almost certainly contributed heavily to their success in becoming the local leader.
But it is equally implausible that the Imperium is not maintaining sovereignty, levying taxes, and all that. Unless trade is just so extensive that levies on interstellar shipments fund it. And that is not in any way established by the setting either. The infrastructure just doesn't suggest that, even if you assume said levies are built into the pricing models we have (which are, after all, simplified to be usable in a game intended to be fun).
IMHO, the "home rule" provisions on the frontiers are allowing them significant planetary navies and tolerating a certain amount of interplanetary conflict that would not be tolerated in the core. The Imperium does not need or want planets in the Core to have their own navies. They do not have the strength or responsiveness to provide naval garrisons to all the worlds in the Marches, so they approve naval militias, essentially.
