I don't think I can really give you the intuition that you seek because I don't think I quite have it yet either. But I think that understanding the relevance of Nigel Higson's comment might help, and I can try to provide some insight. (Full disclosure: most of my understanding of these matters has been heavily influenced by Nigel Higson and John Roe).
My first comment is that the index theorem should be regarded as a statement about K-theory, not as a cohomological formula. Understanding the theorem in this way suppresses many complications (such as the confusing appearance of the Todd class!) and lends itself most readily to generalization. Moreover the K-theory proof of the index theorem parallels the "extrinsic" proof of the Gauss Bonnet theorem, making the result seem a little more natural. The appearance of the Chern character and Todd class are explained in this context by the observations that the Chern character maps K-theory (vector bundles) to cohomology (differential forms) and that the Todd class measures the difference between the Thom isomorphism in K-theory and the Thom isomorphism in cohomology. I unfortunately can't give you any better intuition for the latter statement than what can be obtained by looking at Atiyah and Singer's proof, but in any event my point is that the Todd class arises because we are trying to convert what ought to be a K-theory statement into a cohomological statement, not for a reason that is truly intrinsic to the index theorem.
Before I elaborate on the K-theory proof, I want to comment that there is also a local proof of the index theorem which relies on detailed asymptotic analysis of the heat equation associated to a Dirac operator. This is analogous to certain intrinsic proofs of the Gauss-Bonnet theorem, but according to my understanding the argument doesn't provide the same kind of intuition that the K-theory argument does. The basic strategy of the local argument, as simplified by Getzler, is to invent a symbolic calculus for the Dirac operator which reduces the theorem to a computation with a specific example. This example is a version of the quantum-mechanical harmonic oscillator operator, and a coordinate calculation directly produces the $\hat{A}$ genus (the appropriate "right-hand side" of the index theorem for the Dirac operator). There are some slightly more conceptual versions of this proof, but none that I have seen REALLY explain the geometric meaning of the $\hat{A}$ genus.
So let's look at the K-theory argument. The first step is to observe that the symbol of an elliptic operator gives rise to a class in $K(T^*M)$. If the operator acts on smooth sections of a vector bundle $S$, then its symbol is a map $T^*M \to End(S)$ which is invertible away from the origin; Atiyah's "clutching" construction produces the relevant K-theory class. Second, one constructs an "analytic index" map $K(T^*M) \to \mathbb{Z}$ which sends the symbol class to the index of $D$. The crucial point about the construction of this map is that it is really just a jazzed up version of the basic case where $M = \mathbb{R}^2$, and in that case the analytic index map is the Bott periodicity isomorphism. Third, one constructs a "topological index map" $K(T^*M) \to \mathbb{Z}$ as follows. Choose an embedding $M \to \mathbb{R}^n$ (one must prove later that the choice of embedding doesn't matter) and let $E$ be the normal bundle of the manifold $T^*M$. $E$ is diffeomorphic to a tubular neighborhood $U$ of $T^*M$, so we have a composition
$K(T^* M) \to K(E) \to K(U) \to K(T^*\mathbb{R}^n)$
Here the first map is the Thom isomorphism, the second is induced by the tubular neighborhood diffeomorphism, and the third is induced by inclusion of an open set (i.e. extension of a vector bundle on an open set to a vector bundle on the whole manifold). But K-theory is a homotopy functor, so $K(T^* \mathbb{R}^n) \cong K(\text{point}) = \mathbb{Z}$, and we have obtained our topological index map from $K(T^*M)$ to $\mathbb{Z}$. The last step of the proof is to show that the analytic index map and the topological index map are equal, and here again the basic idea is to invoke Bott periodicity. Note that we expect Bott periodicity to be the relevant tool because it is crucial to the construction of both the analytic and topological index maps - in the topological index map it is hiding in the construction of the Thom isomorphism, which by definition is the product with the Bott element in K-theory.
To recover the cohomological formulation of the index theorem, just apply Chern characters to the composition of K-theory maps which defines the topological index. The K-theory formulation of the index theorem says that if you "plug in" the symbol class then you get out the index, and all squares with K-theory on top and cohomology on the bottom commute except for the "Thom isomorphism square", which introduces the Todd class. So the main challenge is to get an intuitive grasp of the K-theory formulation of the index theorem, and as I hope you can see the main idea is the Bott periodicity theorem.
I hope this helps!
I think you should have a look at the various papers of Louis Boutet de Monvel. But there is actually a construction of star-products on a symplectic manifold which makes use of the index theorem, due to Richard Melrose.
Last but not least, you might also want to have a look at Appendix B of this paper by Engeli and Felder, where they use heat kernal methods while proving a HRR formula for traces of holomorphic differential operators.
Best Answer
The status of this question is OPEN.
This theory has NOT been developed yet.
That being said, the evidence is as compelling as ever, I don't know of any obstructions to making this work, and I'm convinced that there's an awesome theory out there, waiting to be discovered.
Roughly 8 years ago, I wrote an unsuccessful ERC proposal, where I outlined a program. This proposal can be found on my Utrecht website here and here (warning: that website will probably stop existing one year from now, so the links will become broken -- but the linked material will still be there to be found on whatever new website I end up having in the future).
There are small bits and pieces of what one might call progress, which I've made available on my website:
Here's one. In this draft, I take a compact simply connected Lie group $G$ of dimension $d$, and I consider the map $p:G\to \{pt\}$. I construct, geometrically, the $TMF$-pushforward $p_!(1)\in TMF^{-d}(\{pt\})=\pi_d(TMF)$ of the element $1\in TMF^0(G)$ along the map $G\to \{pt\}$.
Here's another one. In this draft, I show that there's a new type of 2-equivariance for $TMF$, where the group of equivariance gets replaced by a fusion category.