"How can our vocal breaks all be within 3 semitones?" Short answer: They can't, because they aren't.
You said it yourself: respective tessitura and/or most comfortable range vary a lot more than just three notes. As for why your singing programs and other sources claim this? Those are average figures. In reality, many singers will fall outside of that narrow range.
There are times when that average is useful, and there are times when it is decidedly useless. Sure, your average man off the street will probably have that sort of range, as most men naturally fall into the baritone range with which the "E4-F#4 break" is very consistent. There are of course tenors and basses who have different breaks in their voices, but on average, and with a high concentration at that point, that's where the male vocal break typically lies.
So if you are writing a (male) solo in your piece and don't know who from your group will be the one to sing it, then you may want to take that into consideration. However, if you're writing for a choir at large, it's a complete mistake to apply that information to the entire male part of the group - one should always work with the best information they have, and the E4-F#4 figure is too general to ignore what kinds of voices each section may have in it.
Another point: the E4-F#4 break mark may be more or less valid in some choirs than in other choirs. A typical high school choir may find itself short on natural tenors and bass voices, and some of the singers in the tenor and bass sections may be more like baritones singing beyond their normal vocal range; thus the statistic may be more accurate. In a grade school choir, this statement is almost certainly going to be false every time, since almost every singer will have a higher range than the average adult male singer. So consider the group, or best yet, consider the individual singers or sections if you can get to know their voices/sounds!