[Sharkû]

Burrahobbit's "theory" has one fatal, if perhaps somewhat elusive flaw – the equation of "songs" with the Ainulindale, and the conclusion that Tom, knowing all such "songs", necessarily has to have full understanding of the Music.
However, the real nature of the Ainulindale is such that we cannot associate it with such degree of certainty and in such bold interpretation with Tom.

The Ainulindale came from the Ainur, and Eru. We do know that the Ainur were given a vision of Eä, their Music put into being; however, we also know this was incomplete. This is one reason why no Ainu, not even Manwe, would have complete knowledge of the Music; the other reason is that Eru brought in themes which are solely his (sc. the Children and others, depending on how meaning is attributed to the Themes, cf. HoME X, I).
These were not then understood by the Ainur, and, as can be assumed with great certainty, Eru might have presented aspects of the Themes not perceived by the listeners, or Eru might still add to the Music afterwards, since that was certainly in his power (cf. Athrabeth).

Burrahobbit circumvened this problem of the Ainur's incomplete knowledge of the Music by making Tom not an Ainu, but Eä itself. As I have argued above long ago (though apparently not convincingly or strikingly enough), Eä is itself only creation and therefore can not only not be assumed to have a full understanding of itself at any time of its existance, but it also cannot be plausible that there could be a perfect simulacrum of itself in itself; in a very peculiar, fixed form (Tom) on top of that.
A painting neither knows how it was painted, nor knows how it looks to the beholder; nor is music able to comprehend itself.

If, however, we are to make Eä a completely different case, not applicable to those comparisons, we are still faced with the difficulty that Eru the One alone and solely possesses full knowledge and understanding of all the the themes of the Music, especially since it is still unfolding, and gradually and perpetually so. Eä, while a finite thing by necessity (cf. Athrabeth), can in its vast boundaries of space and time and drama only be comprehended, more importantly, only be mastered by Eru the Creator alone, as only he can know how and what it is, and how it is going to unfold.

(Furthermore, Tom does not, when tempted by the Ring and the Barrow-Wight, show any sign of corruption, even though we know that Arda, and thus an important part of Eä [even more so with the image prevalent at the time of the writing of the Lord of the Rings that the globe of Arda was pretty much all there was to Eä] was marred. "Arda umarred does not exist." (MT VII). That Tom, if he was to be, represent, or, as a matter of fact, even be within Eä, would have to show at least signs of Marring [though not necessarily at the above mentioned occassions] is evident from the following: "[Melkor] had introduced evil […] into all physical matter of Arda" (Athrabeth commentary). "Latent evil" would have been "roused" by "evil minds" (ibid.) such as the Ring or the Barrow-Wight.
Of course, we cannot say whether Tom is marred, flawed or corrupted or not on the textual basis concerning him, but the texts we have are a firm pointer in the firection that aids my argumentation.)

Therefore, Tom cannot be Eä, and we know Tom is not Eru. Tom cannot be master of the Music, and cannot "know" all songs of the Ainulindale.
On a scientific scale, the absence of Tom from all philosophical or cosmogonical writings of Tolkien only further disproves any interpretation of Tom beyond the old "spirit of the vanishing Oxford countryside", which did not exist until much later ages.

About Tom not as Eä as a whole, but as a mere representation of it, I have said all there is to it in my post near the beginning of this very thread. All parts of Eä, however small or great, evil or glorious are representations of Eä, the end of which is always to add to its greatness as a praise unto its creator.

Tom may be, but Eä is, and Tom is not Eä.