I'm not sure what is proper, but in everyday speaking I've never heard "on" in that context.
While thinking about it a bit, I moved his position:
He was waiting for me on the bottom of the stairs.
Which doesn't sound right, but also doesn't really tell me where he is. I'd almost expect him to be stuck under the staircase like a sticker.
Thinking and researching a bit more, it seems the target indicates which to use, and the rules are pretty vaguely defined. The answers to this question seem to suggest that has to do with the distinction between and surface and an upright target, which I think I agree with (the examples given there might not show that, however, because in that context they also show a difference in intent vs. final resting place)
So, to compliment our "at the top of the stairs" example:
He was waiting for me on the second floor
The expiration date is printed on the top of the jar (I'd expect it to be on the lid)
The expiration date is printed at the top of the jar (I'd expect it to be on the jar itself, near the top)
He was waiting for me on top of the house
He was waiting for me on the roof
Note that at is often used to be more specific about the position/location of the subject.
He was waiting for me at the top of the ladder
He was waiting for me on the ladder.
In the last one, we know he is on the ladder, but we don't know which rung he is at.
What is it called when the preposition is omitted? It's called an indirect object.
That's what an indirect object is. An indirect object is licensed and governed by a verb. When a preposition is involved, the object is a prepositional object or an oblique object.
Oddly enough, the cited dictionary page doesn't manage to offer a clear example of this verb's ditransitive use. The example "he was awarded the Military Cross" is cast in the passive voice. The ditransitive construction is more obvious in the active voice: "They awarded him the Military Cross." The example "a 3.5 per cent pay rise was awarded to staff" is not a ditransitive example, even when cast in the active. "Management awarded a 3.5% raise to the staff" includes a direct object and a prepositional object, but there is no indirect object in sight.
You don't seem to be confused by the ditransitive use of this verb in the active voice. However, passive-voice participles do seem to confuse you.
Such confusion is easy to understand. For most verbs in English, the past-tense forms and the so-called past participle forms are identical. Even the "past participle" label is confusing, given that participles have no tense.
"The Academy awarded the Oscar" looks quite similar to "the actor awarded the Oscar". Without further context, they both can be parsed as clauses in the active voice and past tense. However, "the actor awarded the Oscar has refused to accept it" requires a different parsing. Here, "awarded" is a participle, "awarded the Oscar" is a participial phrase which modifies "actor", and "the actor awarded the Oscar" serves as the complete subject of "has refused to accept it".
You're not the only one experiencing such confusion. It's nearly universal. Sentences like "the actor awarded the Oscar has refused to accept it" are garden-path sentences. It is easy for anyone to assume that "awarded" is a finite verb until the phrase "has refused" is encountered. Since "has refused" is finite and must have a subject, we have no option but to re-parse "awarded" as non-finite and without subject.
The examples that you provided are not garden-path sentences. Each one contains a clue to the participial nature of the verb in question before it is encountered:
The phrase "on the plaque awarded her" has "on", a preposition which licenses an object. The phrase "the plaque" cannot serve as the subject of "awarded" since it serves as (a part of) the object of "on". A similar explanation serves for the prepositional phrases "about the Nobel Prize awarded Ernest Hemingway" and "by the Oscar awarded Michael Radford's Il postino . . .". In "who refused the Oscar awarded him", "the Oscar" is (a part of) the direct object of "refused". In the absence of a subject, there is no clause and there is no reason to assume that the verb's form is finite.
Best Answer
Your rewrite does not have the same meaning as the original. Many English prepositions can be employed without an explicit complement when the complement can be inferred from the context.
Beyond may be used this way, and is used this way in Hemingway's sentence:
You may thus understand beyond here as a sort of 'reduced' preposition phrase acting as a locative modifier on bridgehead. The area the narrator is called upon to explore is the bridgehead itself, not some area beyond the bridgehead: