I too am but a mere graduate student trying to sort through some of these same issues, but I might have some helpful insight. I'll let you be the judge.
The basic idea behind the heat equation proof of the index theorem is to extract the right term in the asymptotic expansion for the heat kernel and then appeal to the McKean - Singer formula. According to my understanding the original strategy for doing this was to realize that the index is a cobordism invariant and thus it would suffice to do enough explicit calculations on generators for the cobordism group until all free parameters are fixed; as it turns out the complex projective spaces are a good choice. That's exactly what was done. I believe - and I really hope someone will correct me if I'm wrong since I haven't gotten my hands dirty myself - that the required calculation really boils down to dealing with the quantum mechanical harmonic oscillator when you work it out for $CP_2$. If this is correct, then the first hint that the quantum mechanical harmonic oscillator is important came from a very fundamental example.
But I think a more analytic answer is also possible. Let's say instead of working with a Dirac operator acting on smooth sections of the spinor bundle you instead just consider the usual scalar Laplacian acting on functions. What happens if you imitate the heat kernel proof in this much less subtle context? You wind up reproving Weyl's asymptotic formula for the eigenvalues of the Laplacian. In essence this calculation amounts to rescaling the spacial variable so that your operator is deformed into the constant coefficient operator obtained by freezing coefficients. The basic idea of the Getzler calculus is to rescale both the spacial variable and the Riemannian metric in a compatible way - this rescaling deforms the Clifford algebra into the exterior algebra (thereby making Clifford multiplication act like an order one operator) and hence the Dirac operator into a polynomial coefficient operator. What polynomial coefficient operator is it? We have reached the limit of my ability to motivate things any further, but the answer is the quantum mechanical harmonic oscillator operator. I of course have no idea whether or not the physical significance of this operator can be accounted for according to a similar rescaling argument.
I should also mention that the quantum mechanical harmonic oscillator makes no obvious appearance in the original global proofs of the index theorem. It does, however, make a non-obvious appearance via Bott Periodicity which can be proven essentially using Mehler's formula. Nigel Higson and Eric Guenter wrote a very readable paper explaining most of the details of this proof entitled something like "K-Theory and Group C* Algebras". You can find it on Nigel's website, www.math.psu.edu/higson.
The last thing I will say is that I found Getzler, Berligne, and Verne to be a pretty tough way to penetrate this material. The style pays off in some of the later material, but I think I would have had a lot of trouble learning the heat kernel proof of the index theorem for the first time from that book. You might try John Roe's book "Elliptic Operators, Topology, and Asymptotic Methods" instead of or as a supplement.
I hope this has been helpful!