On page 558, it is said of the chart on pages 556 and 557 that "[o]ne of the phonetic signs that represent the realizations of a letter comes in brackets when it is less common than the other realizations, or when it is used only by a part of the community in question." I am not, however, sure what yardstick the author is using to determine when a realization fits that description. Would, for example, a mandatory positional variant that's used by less than half of the community, but not necessarily a truly small part of it, get "brackets" (they're parentheses to me)?
In Commonwealth countries, parentheses are called "brackets".
I think the consideration is different for each individual example.
The Italian Jewish pronunciation of yod as an Italian soft "g" is well known among Italian Jews and found in many names. However, I'm guessing that the educated liturgical pronunciation was "y", and maybe the soft "g" was more common in spoken language rendering of Hebrew words. But that's MY PERSONAL SPECULATION.
In the cases of the Arabic-Jewish pronunciation of ת as "ts" (sorry, on mobile and too lazy to make the "s" superscript and the Ashkenazi pronunciation of shuruk/kubutz as "ü", the brackets essentially mean it's a subgroup of the population. The "ts" is a Moroccan feature, and the "ü" is found in a specific region of Ashkenaz (off the top of my head, it's either in or near the transition area between the Central Yiddish and Western Yiddish dialects).
The "ü" in Aramaic-speaking communities is conditioned by the phonemic environment and is probably more or less common depending on dialect.
The other ones ("ü" in Persian and Central Yemenite, "k" in Persian) I don't know enough to say anything.