Almost all divisions had(and still do) non infantry "units" built into them whether it be four battalions of artillery like the US 1st Infantry division had for most of WW2. They ALSO attachments which varied depending on their intended objective. Most of the time these attachments were battlion sized attachments which depending on the army in question is about 1/3 to 1/5 the size of the brigade attachments that you can attach in HOI2/DD/ARM/AoD. From that, one could surmise the the AA and hard attack and soft attack attributes of the ever improving divisions that you can build in the game are a result of abstracting some of these elements, i.e. not only better weapons but better ability to use the right weapons for the right job. The brigade attachment you can add to a division is like adding 3 to 5 battalions so that is where you get your major boost in attributes(major boost all things considered)
The problem with adding multiple brigades to a division, as stated early on, would be that you are adding anywhere between 6,000 and 15,000 people which would basically make it possible to create double strength divisions.
I do agree with Cardus somewhat about the battalion/brigade attachements as follows:
For now perhaps a fun feature would to add multiple battalion level attachments. Bump all the attachments down in stats and cost making them battalion level so you can add 3 or so attachment to each division. you can still achieve the same effect as a tank destroyer brigade attachment but only if you add 3 tank destroyer battalions. But it would give you the flexibility to fine tune your division, such as to add a tank destroyer, a self propelled artillery and a light tank battalion, giving you some of the effects of all three. In short give you the ability to edit the composition of your brigade attachment.