I think this discussion is related to my firm opinion that everything following the verb but before the first e is ill-define outside of a few situations. From jan Sonja's sample sentences, I think she meant this on purpose, that toki pona have weak verbs. Maybe not as weak as , but weak like some papua new guinea or Australonesian languages that have 50, 20 or as few as 3 verbs, and these verbs don't really do much work. (Piraha verbs actually do a lot of work, one of the few places where even Everett admits the language is a remarkably complex and expressive)
S li V [stuff] e DO PP
When the stuff is clearly an adverb, no problems. We see pona, ike, etc in this slot. (somewhere above in this thread I listed all the things I found in the corpus that people had put here when they clearly had an adverb in mind) Well usually no problems, sometimes people gripe that these should be moved to la phrases.
Similarly, when the post verb stuff is "ala", no problems.
When it is anything else, imho, it just isn't defined anywhere in jan Pijes lessons, the jan Sonja lessons.
(A) S li V/P (preposition implying the sorts of things that verbs normally do) noun phrase. This is actually a predicate. mi tawa tomo mi, worried me until I learned about natural languages that extend predication to cover things involving motion and action.
(B) S li V (preposition masquerading as a verb or a verb masquerading as a preposition) (unmarked noun complement) e DO PP. This will continue to worry me until I find a natural language that does this as well.
(C) S li V (verbs of talking, thinking, etc) (unmarked complement, usually what we are talking, thinking about) e DO PP. We rarely see this in practice, the "mi pilin e ni: S" construction is really, really common though, but really verbose.
(D) S li V pi (nouns plus modifiers plus optionally more pi phrases) e DO PP. Who knows what this means? At least in this construction it is clear the post verb stuff is a noun.
I think that these issues mostly have to do with phrases with higher valence (more actors on the stage) and I don't think that has ever been dealt with, and when it has been, the nth actor gets put in a prepositional phrase.
Scenario (B), imho, reminds me of the constructions you see in really old languages, the ones that the children don't really learn to speak well until they're 11, the ones where saying something simple like "She owns a house" is some surprising construction like "It is a house that surrounds her." (and it's all said in one word)
Moving into the land of "jan nasa li wile ante e toki pona", I'd say that constructions B and C should have a marker. It feels like there should be one there. Nothing in toki pona that smacks of a phrase is allowed to be juxtaposed to another phrase without a separating particle. In natural languages, these particles are usually drawn from prepositions, so my if I was feeling avant garde, I'd use one of the free prepositions for the purpose. And once one went down that rabbit hole, I suspect a post verb-pre e-phrase prepositional phrase would come to indicate manner and the post e-phrase would come to represent place. Which would have a interesting correspondence to how Germanic languages seems to care about PMT (place manner time) ordering of prepositional phrases.