Fix a Clifford module bundle $E$ on a compact Riemannian manifold $M$ and let $D_0$ and $D_1$ be two Dirac operators (compatible with the Clifford action). The proof of the Atiyah-Singer index theorem in Heat Kernels and Dirac operators makes use of the fact that in this situation $$\mathrm{ind}(D_0)=\mathrm{ind}(D_1).$$ The proof of the equality is discussed on page 125. However I feel like there is gap in the proof:

We seem so to assume that we can find a family $(D_t)_{t\in [0,1]}$ of Dirac operators such that the function $$[0,1]\ni t\mapsto\mathrm{ind}(D_t)\in\mathbb{R}$$ is continuous. Since the image is contained in $\mathbb{Z}$, it would be constant. In fact there is a very obvious candidate: As explained on page 117 we have that $A=D_1-D_0$ is function-linear, i.e. $A\in \Gamma(M,\mathrm{End}(E))$ and $$D_t=D_0+tA$$ is a Dirac operator for all $t\in[0,1]$. But how do we know that the above-mentioned function is continuous?

The authors suggest to use the McKean-Singer formula, but it was only proven for a Dirac operator which happens to be symmetric w.r.t. some metric. I assume that this is not the case for all Dirac operators (?), so can we avoid using McKean Singer? Or does the latter actually hold for all Dirac operators?