After reading your question, I didn't quite understand what you're asking. Only after reading your own answer did I understand what you're asking and, more importantly, why you're asking what you're asking.
To quote your own answer:
the contextual range for half is more restricted than all and both, both of which are also predeterminers. In short, unlike all and both, there must be a determiner after half.
So, you're comparing half with all and both, which is simply comparing apples with oranges. Firstly, half is a fraction, just as quarter, third, and fifth are.
Secondly, fractions can only function as predeterminers and cannot function as determiners, which means that they can come only before a determiner. So it's only natural that you can't say things like:
*Half Americans disapprove of the President.
Because you don't have any determiner right after the predeterminer!
In contrast, both and all can function as determiners as well as predeterminers. To quote the examples of your own answer:
half those oranges are rotten [predeterminer]
both the children came [predeterminer]
all the men went home and stayed there [predeterminer]
all lions are unpredictable [determiner]
both the dogs are friendly [predeterminer]
both dogs are friendly [determiner]
*half people arrived [predeterminer]
*half trains are always late in my country [predeterminer]
So you just have to know that half is a fraction whereas all and both are not, and therefore that half functions differently from all and both.