The 'go' test:
He went in.
He went across.
He went home.
He went bananas.
The 'stay' test:
He stayed in.
*/? He stayed across.
He stayed home. (US)
He stayed put.
The 'put' test:
Put it in.
*Put it across. (non-idiomatic)
*Put it home.
Put it concisely.
The 'place' test:
??Place it in.
*Place it across.
*Place it home.
Place it carefully.
The 'position' test:
*He positioned it in.
*He positioned it across.
*He positioned it home.
(He positioned it just here.)
He positioned it carefully.
The 'send' test:
She sent him in.
She sent him across.
She sent him home.
She sent him immediately.
What does this prove about the prepositionality of in, across and home?
And, from your second link, a comment with which I wholeheartedly agree:
My counter-argument to Geoff's argument that there are objectless prepositions, e.g., the under in 'the ship went right under' is that in this case the object is understood from context. It may be 'the waves' or 'the sea' or even 'the water'. It's not there but it's understood; it's elided. In the case of the chicks that headed bush, what is understood is the preposition. In this case I agree that bush is not an adverb, it's the object of an understood and elided preposition, to. The definite article is also elided just because 'the chicks headed the bush' just doesn't sound right. The whole phrase, 'under the waves' or 'to the bush' as acting as an adverb, which is why the trads will try to make bush be an adverb, but I say that it is the object of an understood preposition, the whole prepositional phrase acting as an adverb.
When there is an elision in an accepted construction (and as we know, this is often the case), it becomes nonsensical to analyse the elided construction syntactically as it now stands, using similar-looking constructions that it now mimics, but without reference to what the full, also grammatical, version was. We end up with analyses such as 'lead is ditransitive in he led them a merry dance', and 'take is ditransitive in he took the dog a walk'.
Your first link claims wrongly that 'prepositions can typically function as complements of be':
• The dinner is afterwards (after the dance).
• Good things are ahead (in front of us).
• The shortest way is east (to the east).
• They’re here now (at the conference).
• Tell me when you’re home (at school).
It's the prepositional phrases that 'can typically function as complements
of be' (eg to the east not to; at the conference / school not at).
As in the first quote here, the whole prepositional phrase (whether given fully or elided, leaving the elided elements to be mentally reclaimed) is acting as an adverb. It is a matter of opinion whether the then preposition is still a preposition (or then noun - eg bush in went bush - is still a noun); however, where there is obviously no elision, as in they are here, inventing rules to support inclusion in the preposition class is unsatisfactory. From a distributional argument, here behaves quite like an adjective or possessive pronoun (or adverb) - it depends on which tests you choose to avoid. Prepositions have traditionally been defined semantically (describing spatial and temporal relationships) and then, by syntactic extension, in the construction of other similar structures (prepositional phrases) not directly reflecting semantic relationships. And yes, this traditional model is itself not without difficulties.
For instance, There's a debate about whether we should take the train or not. doesn't have a typical noun phrase / clause following the preposition about.
But it does have a substantive, a noun-substitute, as shown by:
Whether we should take the train or not isn't your problem.
Also, I didn’t hear until later.
Later will do.