Why does CGEL call "home", "abroad", "here", and "there" prepositions instead of adverbs like dictionaries say?

In an article published on Geoffrey Pullum's website, "The Theoretical Orientation of The Cambridge Grammar of the English Language," Pullum and Rodney Huddleston provide a succinct defense of their redefinition of preposition. In short, Pullum and Huddleston find that the conventional definition of prepositions arbitrarily groups similar usages under other categories of speech (like adverb and conjunction); it makes sense for them to consider words applied to all of these uses as prepositions.

Consider this extended excerpt, where the authors present the way that a similarly-appearing word may be classified in three different ways with only slight changes:

It is also clear that uncontroversial prepositions like in, up, down, over, through, etc., are sometimes not followed by a noun (or NP) or anything at all:

• Soon they went in the house.

• Soon they went in.

• He came running up the street.

• He came running up.

• Does this hole go right through the wall?

• Does this hole go right through?

• They like to run around the yard.

• They like to run around.

To take account of such facts, traditional grammar posits that a substantial subset of the prepositions have homophonous and virtually synonymous doppelgangers belonging to other categories. Thus they recognize the word down as a preposition in He fell down the steps (it is followed by the NP the stairs) but not in He fell down on the steps. In the latter, since there is no NP following it, down has to be an adverb (it does, after all, modify the verb fell). Another (overlapping) subset are alleged to have doppelgangers in the ‘subordinating conjunction’ category: in before her court appearance the word before is acknowledged as a preposition, but in before she appeared in court, where what follows before is a declarative content clause, it is claimed not to be.

There is no semantic, morphological, or phonological support for having three words spelled before. (emphasis mine on the last sentence)

The second of each example shows that "uncontroversial prepositons" already can appear in positions where they do not pre-pose or appear before any NP (noun phrase) at all. Why are other words, otherwise similar in how they appear in a sentence, grouped as adverbs or conjunctions instead?

To your question, I find the last example, "I like to run around," especially useful, because that sounds fairly close to the question you are raising with home. Around and home both have directional senses. If around here is a preposition, and you accept that not all prepositions have to precede NP, then why can't home function in a similar way as a preposition that doesn't have to precede an NP? So in I like to run home, home functions as an intransitive preposition (a preposition that takes no NP complement) rather than an adverb.

There is more to why Huddleston and Pullum categorize words as they do, for which I recommend a more careful reading of CGEL. Huddleston and Pullum are only one of several opinions on home - the traditional grammarians they are responding to still exist, for example.