Bill;
I know the original intent of battle patches was to provide instant recognition in the field - mostly for infantrymen in World War One - which is why the patches were originally intended to be sewn on the back of tunic collars. I believe some British units did in fact adopt that practice. Canadian units wore large geometric shapes on both arms, as well as on steel helmets, so that when advancing over featureless terrain, one could tell what battalion a man was from.
During the Second World War, I would submit that the need for Corps Troops to be distinguished from Army troops "at a glance" simply did not exist. Hence the similiraties between corps and div patches. Further, I don‘t see that formation patches of any kind were a necessity in the field, even by infantrymen, by 1939. Strome Galloway recounted that one of his troops remarked in Sicily that they looked like a field of poppies on the move, with their Old Red Patches proudly sewn on their sleeves.
In reality, though, the closer you got to the front line, the more often you saw all badges removed - especially badges of rank, but also the brightly coloured regimental and divisional patches. Fear of death by sniper fire outweighed any pride the troops may have had in a couple of pips on their shoulder, or a battle patch on their sleeve. It was universal; even German Army officers took steps to look more like enlisted men, covering their shoulder straps, wearing enlisted men‘s uniforms, and colouring their brown leather gear (before the war a symbol of authority) black.
Clothing wore out and was replaced with new clothing (sans badges), and many soldiers were ordered to take down their badges (ie when the Canadians in Italy were moved to the coast for Goldflake, the repatriation to First Canadian Army, they removed all formation and unit insignia for security reasons).
That was probably a consideration when designed the army patch - there was no need to have a hexagon or octagon or something equally dissimilar to a rectangle or a diamond. Army troops would not be likely to be in the front line, and even if they were, no one would be relying on the patches on their sleeves to provide them necessary information (excepting perhaps German snipers (if they were alive) or German intelligence officers (if they got killed or captured))
If you look at the div patches for 6, 7 and 8 divisions, you can see they used a similar shape to the other divs, but combined the colours. The Canadian Army Pacific Force patch also combined the colours of all the div patches of the 5 overseas divs (and added black - the original colour of the 3rd Div patch in World War One). Not entirely imaginative. Perhaps you can say that the First Canadian Army patch, and its designer, was part of that same trend of unimaginativeness. Also called, in the Army, "uniformity."
Just my random thoughts, anyway.