"I tell you, monks, that this — the in-&-out breath — is classed as a body among bodies, which is why the monk on that occasion remains focused on the body in & of itself [not the flesh body or mental body]."*
Although we can just treat the whole physical body (including both the breath body and the flesh body) and the mental body as an entirety, the sentence "experiencing the whole body I breathe in/out" in the sutta itself seems to mean following the entire breath (a very good way to concentrate the mind), instead of pervading the breath sensation to the whole (flesh) body which doesn't seem to be remaining "focused on the body (breath) in & itself".
Sounds to me as if the statement
"this — the in-&-out breath — is classed as a body among bodies, which is why the monk on that occasion remains focused on the body in & of itself" is showing how the breath is a thing tied up with the body (a bodily fabrication), which is why when we develop mindfulness of breathing we develop mindfulness of the physical body. If you watch the breath closely you'll notice that the process of breathing has a subtle impact on the entire body. Which is why mindfulness of short/long/in/out breathing (that is, breathing in general) transitions well into all-around bodily awareness [3rd step], which then transitions well into a more intimate knowledge of what's going on in the body, and pacifying those processes (that is, "calming bodily fabrication", (including the breathing), [4th step]) for the sake of building a stable platform for building deeper concentration.
However I do agree that the Anapanasati sutta doesn't say anything quite like "spreading breath energy" which some teachers recommend, but I think it's probably just another tool for reaching the same end.
It seems to me that “Or his mindfulness that 'There is a body' is maintained to the extent of knowledge & remembrance" in MN 10 also refers to the in-&-out breath as far as Mindfulness of breathing is done.
Except that the Satipatthana Sutta goes on to explain "body" fairly unambiguously in terms of the physical body. I won't quote it, because it's long, but it's right after the portion on breathing. Which relates back to another thing, that this first quote* from MN 118 is from 1 of 4 parts showing how the tetrads of anapanasati relate to the 4 satipatthana. I believe that since the 1st satipatthana appears to be referring to the usual physical body, it can be inferred that the 1st tetrad of anapanasati also deals with the physical body, otherwise they really wouldn't be compared in that way.
Double-edit: Oh, I must have read too quickly, I now noticed that you said "as far as Mindfulness of breathing is done". I guess imo saying that in the context of anapanasati "body" as a satipatthana takes on another meaning of "breath body", excluding "body" in the usual sense, doesn't quite make sense.
In short I believe what's going on here is that because the breath is something tied up within the body, it is a good tool for developing mindfulness of the body. I also kind of suspect that being aware of the whole duration of the breath is implied in the first 2 steps of anapanasati anyway.
Edit: I just remembered that this is the Pali forum, whoops. Eh, no harm done.