In the introductory chaper of Renormalization and Effective Field Theory, Kevin Costello defines a propagator $P$ as the integral kernel for the Laplace operator $D$ in a Riemannian spacetime $M$. In order to define a "length-scale version of the renormalization group flow", he provides a formula for this propagator in terms of the heat kernel $K_\tau$ of the Laplace operator: $$P(x,y):=\int_0^\infty d\tau K_\tau(x,y).\tag{1}$$ Using the interpretation of the heat kernel as a transition probability, one may then rigorously write (expanding upon equation 1) $$P(x,y)=\int_0^\infty d\tau \int_{\gamma:[0,\tau]\to M\\\gamma(0)=x\\\gamma(\tau)=y}\exp(-\int_0^\tau \|d\gamma\|^2).\tag{2}$$ My question regards the interpretation of this "path integral" above involving $\gamma$, as something is confusing to me. Since Costello defines $\tau$ to be the proper time of the worldline $\gamma$ as measured by the Riemannian metric, it occurred to me that the integrand in equation (2) vanishes for $\tau < \text{dist}(x,y)$, as there will not exist paths with lengths smaller than that of the geodesic.

Therefore, we can write, more subtly, $$P(x,y)=\int_{\text{dist}(x,y)}^\infty d\tau \int_{\gamma:[0,\tau]\to M\\\gamma(0)=x\\\gamma(\tau)=y}\exp(-\int_0^\tau \|d\gamma\|^2).\tag{2}$$ Look how, naturally, a length-scale cutoff emerges. Clearly, I have made a serious conceptual error. Which is it?