The tube is called a balance tube because it is an attempt to balance or equalize the vacuum seen by each carburetor. This vacuum might (see footnote) differ on each side due to unequal ring, bore, or valve guide wear. If the vacuum is unequal, then the fuel draft will be unequal, causing the engine to idle erratically, which in turn will create an imbalanced vacuum, creating a greater fuel draft imbalance. etc.
Conditions such as unequal wear mentioned above, cannot be rectified with an external crosstube, hence this tube is merely an attempt to rectify the imbalance caused by such wear. In short, it helps. In theory, if an engine is well built to specification on each cylinder, the balance tube should be unnecessary, although in practice, it may still help.
If the balance tube mainly compensates for the vacuum difference due to unequal wear, then the ideal tube diameter and length cannot be rigidly specified for any particular engine condition. Moreover, even with equal wear on each cylinder, should there be an ideal balance tube diameter, it will require adjustment as wear progresses.
Footnote: the main cause of vacuum imbalance between cylinders is usually due to the carbs themselves being whacked out by a novice or inept carb tuner. The obvious causes are leaky slides, unequal slide cut-aways, unequal idle jet sizes, and most likely ( as I have usually found when helping out someone who claimed his carbs were "hopelessly whacked out"), the engine was running on one carb due to a combination of faulty throttle stop and idle jet adjustment.
Slick