The Americans went through a number of different structures but their WW2 armored division basically had 6 x tank battalions (4 x medium and 2 x light) and 3 x mechanized infantry bns. They were generally divided into two Combat Commands (A and B - albeit that a third Combat Command R[eserve] could also be formed.
Quibbling...the 1943 ("heavy") organization had 6 + 3; the 1944 ("light") organization was 3 + 3 and only the 2A and 3A retained the "heavy" structure. There were two primary formation HQs (CCA and CCB) to which battalions were assigned (theoretically task-organized, but often retaining affiliations for long periods), with CCR sometimes being used. So in the 1944-45 time frame when the US was doing most of its useful European land fighting, the establishments were mostly at 1:1, totalling 7 manoeuvre battalions if the recon battalion is included. Details aside, this does suggest a leaner (fewer manoeuvre battalions) model.
The German WW2 Panzer division settled into a structure with a panzer brigade (2 x tank regiments of 3 x tank coys each) and a motorized infantry brigade (2 x motorized infantry regiments of 3 x rifle and 1 x MG coy each).
Quibbling again (mainly because I consider the 1944-45 time frame the most important for gauging the learning of WW II armies), the two regiments were of two battalions each (one mech, three motor). So 1:2, but the anti-tank battalion, usually StuG-equipped in a panzer division, could be used as the basis of a third grouping with armour. But, again, adding in the aufkl battalion, a total of 8 manoeuvre battalions; less than the triangular "9" template and supporting your argument for smaller divisions since there were 6 tank or infantry. Militating against that is the observation that the Waffen SS divisions were established with a couple more infantry battalions. I take that not just as some kind of irrational one-upmanship, but as evidence for "you can never have enough infantry", even in armoured formations.
(1944-45 Soviet tank corps were considerably different, and although on first glance look tank-heavy on a line chart their smaller tank companies arguably put them somewhat below 1:1 - fewer tanks - while their mech corps were at about 1:2.)
Key to all this is fire support, for which there are not many examples of western armies that did not establish field or light firing battalions on the scale of one per lower manoeuvre formation in their divisions.
To extend: 6 tank or infantry battalions for US and Germany and 7 for Commonwealth. A question I find important is: 2 or 3 lower formation HQs to employ them tactically?
Also, a point I keep putting out because part of the Canadian lamentation is whether we could ever afford the tanks for an "armoured division", two battalions of a "44" establishment is probably enough for an effective division that is something other than a pure mech infantry division. Overall, a battalion ratio closer to 1:2 than 1:1, with 3 lower formation HQs, would be my starting point for modeling and experimentation.
Now, do those lower HQs deal with their covering force-type and other traditional recce/cavalry ("light") tasks out of their manoeuvre assets, or do they need an established element ("cavalry"), and does a list of probable tasks for any given phase amount to more than a sub-unit can handle? How would a task-organized brigade of two tank/infantry battalions and a cavalry battalion, supported by a CS fire control and planning element (irrespective of whether it "owns" any firing platforms), perform on offence and defence?
