I do have one question is the presence of the borrowing operation the only significant difference between Westergaardian and Schenkerian theory?

The short answer is: definitely not. The long answer (a discussion of the relationship between Schenkerian and Westergaardian theory) is too long for this comment, but is something I plan to write about in the future. For now, be it noted simply that the two theories are quite distinct (for all that Westergaardian theory owes to Schenker as a predecessor) -- and, in particular, a criticism of Schenker can by no means necessarily be taken as a criticism of Westergaard, and vice-versa (see below).

For me the most distinctive advantage of Westergaardian analyses is that it respects the fact that notes do not have to "line up" according to a certain chord structure. Notes that are sounding at the same time may be performing different functions, whereas harmonic theory dictates that notes sounding at the same time are usually "part of a chord" which is performing some harmonic function.

The way I like to put it is that in Westergaardian theory, the function of a note is defined by its relationship to other notes in its line (and to the local tonic, of course), and not by its relationship to the "root" of the "chord" to which it belongs (as in harmonic theory).

A corollary of this seems to be that Harmonic analyses work fine when the notes do consistently line up according to their function

If by "work fine" you mean that it is in fact possible to identify the "appropriate" Roman numerals to assign in such cases, sure, I'll give you that. But what is such an "analysis" telling you? Taken literally, it means that you should understand the notes in the passage in terms of the indicated progression of "roots". Which, in turn, implies that in order to hear the passage in your head, you should first, according to the analyst, imagine the succession of roots (which often, indeed typically, move by skip), and only then imagine the other notes by relating them to the roots -- with the connection of notes in such a way as to form lines being a further, third step. To me, this is self-evidently a preposterously circuitous procedure when compared with the alternative of imagining lines as the fundamental construct, within which notes move by step -- without any notion of "roots" entering at all.

Having said that, my biggest worry with Westergaardian theory is that it is almost too powerful. Whereas Harmonic theory constrains you to producing notes that do sound in some sense tonal (for a very powerful example of this see here)

I am as profoundly unimpressed with that "demonstration" as I am with that whole book and its author -- of which, I must say, this example is entirely characteristic, in its exclusive obsession with the most superficial aspects of musical hearing and near-total amputation of the (much deeper) musical phenomena that I care most about and find most interesting. As far as I am concerned, there is no aesthetic difference between any of the passages (a) through (d) for the simple reason that all four of them are too short to possess much of any aesthetic characteristics in the first place: they all consist of three bars of four chords each. They are stylistically distinct, I suppose (though not actually very much, in the scheme of things), but any of them could be continued into something interesting or something less than interesting. One thing, however, is certain: if any of them were to be continued in the way they were generated (i.e. at random), the result would be nothing short of awful -- and equally so in all four cases.

The essence of musical composition -- at least its most fundamental and "elusive" aspect -- has to do with projecting coherent (i.e. recognizably human-designed) gestures over long time spans. (How long "long" is depends on context: even if you're writing a ten-second piece, you will want to carefully design its global structure.) The point being that multileveled thinking -- control of all the various degrees of locality and globality and their interrelationships -- is at the core of this art form. For that, you need a hierarchical or "reductive" theory (the very thing that our author explicitly says he doesn't want, even claiming that to hear this way is beyond human cognitive capacities -- I'm not making this up, see the last part of Chapter 7), which harmonic theory isn't. To be impressed by the difference between (a) and (d) -- as readers are apparently expected to be -- is to miss most of the point of what music is about.

Westergaardian theory seems to allow you to do almost anything whether it sounds musical or not.

Not as Westergaard sees it (see e.g. the last paragraph of p. 294 of ITT). I actually think he's wrong about this, and that the theory should allow any note to happen at any time; the theory after all is supposed to constrain analytical choices, not compositional ones. A composer can write anything, and the question for the theorist or analyst is how a given listener understands what the composer writes.

While it is very easy to come up with a Westergaardian analysis, it is very difficult for me to understand why someone who had a certain framework in mind would have performed the operations that would have led them to the music in its actual form. The main culprits of this seem to be anticipatory notes and borrowing.

It's hard to address this without a specific example to discuss.

One more thing: Have you read "why I am not a Schenkerian" by Lodewidjk Muns? Here is the link: http://lmuns.home.xs4all.nl/WhyIamNotaSchenkerian.pdf

That's not an interesting critique of Schenker, let alone Westergaard (who is not mentioned or cited even once). It basically goes like this:

(1) Schenker did not adhere to rigorous philosophical standards in his rhetoric.

(2) I disagree with (or don't understand) some of Schenker's analyses and those of his disciples.

(3) Therefore, harmonic theory is correct.

I'll also note that while some of the criticisms of Schenker are legitimate (if boring), others are completely wrong (e.g. the idea that the highest structural dominant is necessarily the final one).

Here is my attempt at fleshing out a more specific example: http://i.imgur.com/ruEYlhD.png I can't figure out how to generate those using using Westergaardian theory

Use octave transfer (ITT sec. 7.7).