The original sentence is grammatically correct, as is yours. There are several possibilities as to the why of the structural choice.
In highly conjugated and declined languages, such as Latin, a word's grammatical role is usually clear based on its form, and placement is used for emphasis. Placing words first or last in a sentence, regardless of whether it is the subject, verb or object, tends to indicate its importance to the idea.
Because English is not highly conjugated or declined, placement is often the principal indicator of a term's grammatical use in the sentence. The following sentences do not mean the same thing:
John hit Peter.
Peter hit John
However, there are structures in English that allow reorganization of position without making the grammatical meaning ambiguous. In those cases, position can be used for emphasis. The predicate nominative is one of those types of structures.
In the following sentence, there does not seem to be any significant emphasis on any particular word or term.
There are probably two reasons for delaying finishing it.
The original sentence has a somewhat less common structure. By itself that creates a bit of emphasis on the whole sentence.
The reasons for delaying finishing it are probably two.
Additionally, placing reasons at the beginning of the sentence seems to place a slight emphasis on the word, both because of its initial placement, and because it is in a location where it usually would not be found. If it were spoken, one can imagine a rising inflection on the first syllable of reasons.
Further, alternative structures are often used just to make a sentence seem a bit less mundane and to avoid repetitive, sing-song pacing.
"[...] questionable and unsafe places to sleep had plunged him into a constant alertness, one that had only been reinforced when [...]"
Grammatical, but weakly written.
*EDIT - questionable = unsafe. you risk watering down the sense of danger. had plunged - why past perfect? also, hyperbole? Constant alertness - if you are not continuously [in focus], could you be said to be alert? Been reinforced - passive is OK, but why? (and, lose tempo.)
"After what he perceived to be an uncertain amount of time [...]"
Grammatical.
"The noise paused in front of the door"
Grammatically correct, semantically invalid, for the reasons you stated. Prefer "the noise stopped"
can a sound "gain proximity" ? (i.e getting closer).
Prefer "draw nearer" or, simply, "intensified," or the like
"[...] he caught a glimpse of a small group of guards facing the door. Somewhere in his mind, he noted their unusual numbers, though it barely sparked any curiosity in him"
Numbers --> Number, singular: a large (unusually large) number
Prefer switch "it" with "this" to reinforce predicate.
"his thoughts had been slowed down to a halt by the bland nature the days had acquired."
Grammatical, but weakly written.
*EDIT - 1) idiomatic and uninspired. This is no place to make suggestions, but as you have asked, why not: "Time now ran together and his thinking, not at all. Yesterday, today, tomorrow: mere words, no longer concepts, no longer tied to anything, any anchor, of sense and sensing. His Being bled through the fifth dimension listlessly..."
Best Answer
Your reasoning (You cannot treat "both of us" as a list of two alternatives because 'both of' explicitly means they are not alternatives) is correct.
There is no syntactical explanation. "One of (the) both of us" is as grammatically sound as "Colourless green ideas sleep furiously". It is the semantic content of the words that makes both them unacceptable.