That is Imperial propoganda.
The Imperium decides the interstellar trade policy of member worlds.
Interstellar policy is a matter for the Imperial Bureaucracy to manage.
Domestic policy within star systems is a local matter controlled by local governments.
The Imperium has a subsector level of government that ensures worlds do as they are told or else.
Who's spreading propaganda now?
There are numerous examples of Imperial intervention into world affairs.
Yes ... but ... most of those examples have to do with either domestic policies bent on self-destruction (nuclear war between rival factions) that can lay waste to planets and populations ... or are examples where megacorporations have "run amok" under their remits on planets, resulting in unsustainable resource extraction at the expense of planetary environment health (basically strip mining the place and leaving the spoil waste lying around to rot for someone else to clean up). Another example would be when a particular world/star system becomes a haven for pirates or other criminal elements that threaten freedom of navigation (and commerce) in the surrounding region.
When you've got over 10,000 star systems to choose from ... there are going to be examples of "Waste, Fraud and Abuse" in the systems and policy apparatus no matter what you do. The difference is whether those situations are the NORM and only to be expected ... or are they the exceptions that prove the rule (where greed outpaced wisdom)?
The control of trade ensures that TL15 industrial and high population worlds manufacture the bulk of goods and the lower TL worlds are effectively impoverished.
You say "control of trade" ... where anyone else can see "obvious competitive advantage" ... and attribute nefarious motives to the practice.
Even in the far future, laws of supply & demand along with competitive advantages "still work" at interstellar scales.
The Imperium does just that when it suits them, again canon examples abound.
Control of a world's interstellar economy gives the Imperium effective control of that world.
Does it?

I think you are (desperately) overstating your case.
INFLUENCE over world affairs, I will grant ... because it is impossible to have a trade relationship with foreign entities without a measure of influence being granted (in both directions, but the measure of influence can certainly be unequal, depending on the relationship).
CONTROL is something that is flatly wrong.
Not to drag something that belongs in The Pit into this discussion, but that's akin to saying that because Canada has a trading relationship with the US, ipso facto the US "controls" Canada. We have a real world example of how that is NOT THE CASE unfolding right now on the world stage.
They could do so with ease, there is a department for education for each subsector which has money to distribute to worlds. I wonder what they insist is on the curriculum...
Again, not to drag The Pit into this discussion ... but are you asserting that there is some kind of Imperial "Common Core" curriculum that MUST be taught to every child in every school in the Third Imperium? That this "Common Core" is IMPOSED, top down, onto EVERY world and citizen throughout the Third Imperium?
And yet those improvements are never made...
Which kind of cuts against the notion that the Third Imperium "controls" worlds (via interstellar trade channels?) if even the "improvements" the Third Imperium MIGHT want to make "never happen" as you assert.
But the Imperium could allow the worlds of the subsector of Rhylanor the right to elect their subsector government. They choose not to use that model.
LOOK at the UWP Government codes.
LBB3.81, p11:
By my count (and feel free to check my math on this one) ... there are only TWO government types in that listing that yield "popular elections" for rule over world governments.
An argument can be made that Code: 7 can also yield "popular elections" of leadership, but only for a subset of the world's population (because multiple governing factions rather than a single unified world government).
And just for reference, Government Code: 2 is only possible (typically) up to Population: 7.
Government Code: 4 is only possible (typically) up to Population: 9.
So if you're wondering WHY the star systems in the Rhylanor subsector "don't elect their subsector government" ... the simplest answer is
because elected governing (of any variety) is a vast minority of government types for the worlds in the Rhylanor subsector.
A quick look at LBB S3, p30 shows that only THREE of the 32 star systems in the Rhylanor subsector have Government Code: 4 ... and only ONE of those 32 star systems has Government Code: 2.
EVERY other star system in the Rhylanor subsector has a non-elected form of (world) government.
Now explain for the class why 4 Non-industrial worlds with "modest" populations should get to dictate the "rules of subsector government" to the 131.1 BILLION people inhabiting the other 28 star systems within the subsector.
Even for someone who zealously believes in egalitarian democracy, 4 out of 32 star systems is a super-minority constituency for the proposition that subsector governing authorities should be determined by popular election of representation.
They choose not to use that model.
Something something ... vulnerable to the whims of demogoguery ... popularity contests aren't the best way to select for expertise in governing ... blah blah blah ... you know the drill.
If I "grant" you it means I control you, as I can rescind my "grant".
So if I "grant" that you might have said something that is correct, does that mean that I now control you?
I'm detecting a flaw in your logic here.
