I think you are confused that the upper and lower limits of the integral are the same, so it must be zero. Think about non-relativistic QM, does the amplitude for a particle to transition from $x$ to $x$ equal 0?
The limits written in the path integral are not values that you have to subtract off like in single variable definite integration. Think about non-relativistic QM: The integration is being done over a large number of intermediate $x_i$ variables $i=1$ to $N$. The ends of these paths are kept fixed while the intermediate variables are integrated from $-\infty$ to $\infty$, which are the actual limits of the definite integration being done.
More generally, how do we implement boundary conditions in the path integral?
In QFT, we don't usually do this path integral (which is ill-defined) . We just write it formally, use its formal properties to differentiate it wrt $J$ and derive a perturbation series for the correlation function.