I think the Imperium suffers from two major weaknesses. First, it typically acts reactively instead of proactively. Second, it doesn't seem plan on the macro level that much. Or perhaps it simply doesn't do so that well.
Where the first is concerned, this seems to be a result of the sheer bulk of the Imperium. It's so large, and has so many enemies, and suffers from so many problems, that it can't possibly address all the issues it must. A prime example of this is the Imperial Guard fluff regarding armies arriving years after the end of a war, or the Guard ordering the sanction of a general they didn't know was dead decades before their proclamation.
Where the second is concerned, that too seems to have a lot to do with the Imperium's size. The Imperium is plagued by so many problems, and most efforts seem to go toward addressing immediate issues rather than root causes or upcoming problems. Hive worlds only grow in population, even though they require obscene amounts of resources from surrounding worlds and systems. Did the Imperium plan for such inefficient worlds to exist? Judging by the description of the hives themselves, I would guess not.
On the flip side, though, we do have examples of things the Imperium does right. Forge Worlds, for instance, are a strength of the Imperium. Their industrial capacity is absolutely necessary. Sure, they're a cautionary tale of unchecked industrial damage, but only in the real world - not in a fictional dystopia where Humanity is opposed by supernatural forces led by gods that thirst for our blood.
How would I run things? Well, I think the answer was already out there, but the High Lords kind of forgot the obvious.
Think about the Great Crusade for a second. After the unification of Terra and Mars, and following what was probably decades (at least) or even centuries of preparation, the Emperor launched an obscene amount of forces against the Galaxy. Those forces only grew after more planets (and Forge Worlds, especially) were recovered, but still - those two planets generated a stupendous amount of military materiel, starships, and warriors. Now, was Mars the pinnacle of a Forge World? Sure, and I have no doubt it played a role on the quantity of materiel that was produced. Hold on to this thought for a second.
Let's also consider what we know about the Space Marines. In their current incarnation, many of their Chapters are incredibly inefficient. Much has been said about their selectivity, and how difficult it is to find a proper recruit... but think on four of the First Founding Chapters that we know of:
1. The Ultramarines hold six-seven inhabited planets, with a population numbering in the hundreds of millions (if not billions) and an infrastructure specifically geared toward sustaining a Chapter of Space Marines.
2. The Imperial Fists have Necromunda, which contains countless hives and a population we can only guess at. They are content to recruit from any of the planet's social strata, to include criminals and the scum of society.
3. The Blood Angels have Baal Secundus, an irradiated wasteland with an insignificant population of irradiated, comparatively primitive, nomadic tribesmen.
4. The Dark Angels bounce from planet to planet, recruiting from feral cultures. On at least one occasion, they got two viable recruits (and it should have been three) from a couple dozen teenagers.
What does this tell us? The selectiveness Space Marine Chapters have with their recruits probably has more to do with the fact that they are bound by their paranoid masters to keep their numbers at a specific level. Given this, it wouldn't be unusual for a group that considered themselves elite to impose incredible standards... even if said standards weren't really necessary. In all actuality, it's the psycho- and hypnotic indoctrination, their mental training, and their superhuman organs that make Space Marines what they are: not the fact that they were King of the Hunters on Jungle World X.
So, my solution:
I would have (because it's too late now) leaned forward about a millennium or so and established a number of what I would call "Crusade Systems", from scratch. The intent for each of these systems would be to have them operate independently from the rest of the Imperium insofar as they would not require resource assistance past a certain point of development, nor would they contribute (Tithe level Adeptus Non).
Each of these Crusade Systems would have, at a minimum, a Forge World and what would functionally be a Fortress World. Once the infrastructure was in place (the longest part of the process), each System would have, say, a five hundred year timeline. One world would crank out materiel and starships; the other would breed warriors. Each First Founding Chapter's Geneseed would be used to create a single Chapter within the system. The intent would be for each Crusade System to kick off what Terra and Mars did, albeit in miniature form, at the end of their respective five hundred year (or whatever) timeline. Nine different crusade fleets would sail out, each headed by a Chapter of the Adeptus Astartes (which would obviously be a crusading Chapter), each with its own fleet and Imperial Guard Army, each with its own Mechanicus contingent, each with its own Inquisitorial representation.
So what you would basically have is a recurring calendar of Crusades being launched from each of these systems at the end of every cycle. Rather than the ad-hoc system of launching Crusades against a specific threat or location, these systems would churn out Crusades to seek out and destroy the Imperium's enemies independent from whatever the rest of the Adeptus Astartes, Imperial Guard, and Imperial Navy did.
I don't know how realistic it would be, but I would want no less than roughly a hundred such systems established - which could mean as many as a thousand Crusades above and beyond whatever the Imperium normally organizes every few centuries. Each of those Crusades would have the advantage of not weakening the Imperium in terms of resources or manpower. Each of them would also benefit from a force structure that is still divided in terms of authority (per the Codex Astartes) and overseen by the Holy Ordos, but has been growing, planning, and training specifically to fight together.
The ultimate intent for this system would be to take the weight off the Imperium enough for me to address the inefficiency inherent in a great deal of the Imperium's worlds, systems, and even sectors - over the long term, of course. Given some flexibility, and having had some of the tithe burden removed from them, certain planets might re-address the way they utilize their manpower (less Necromunda, more Thracian Primaris), becoming more efficient and independent. This might in turn make the defense and infrastructure of the Imperium more bearable.