ChatGPT-Calculation:
$$\vec{\nabla}^2 \frac{1}{r} = \frac{1}{r^2} \frac{\partial}{\partial r} \left( r^2 \frac{\partial}{\partial r} \frac{1}{r} \right) = \frac{1}{r^2} \frac{\partial}{\partial r} \left( r \cdot \frac{-1}{r^2} \right) = \frac{1}{r^2} \frac{\partial}{\partial r} \left( \frac{-1}{r} \right) = \frac{1}{r^2} \left( \frac{1}{r^2} \right) = \frac{1}{r^4}$$
My Calculation: $$\vec{\nabla}^2 \frac{1}{r} = \frac{1}{r^2} \frac{\partial}{\partial r} \left( r^2 \frac{\partial}{\partial r} \frac{1}{r} \right) = \frac{1}{r^2} \frac{\partial}{\partial r} \left( r^2 \cdot \frac{-1}{r^2} \right) = \frac{1}{r^2} \frac{\partial}{\partial r} \left( -1 \right) = \frac{1}{r^2} \left(0 \right) = 0$$
Did I make a mistake in the calculation?
Without any calculation, you should immediately see that the ChatGPT answer must be wrong simply on dimensional grounds. Assuming that $r$ has the dimension of a length $L$, the left-hand side of your first "equation" has dimension $L^{-3}$, whereas the right-hand-side has dimension $L^{-4}$. On the other hand, the correct result $\Delta \frac{1}{r}= - 4 \pi \delta^{(3)}(\vec{x})$ is dimensionally correct as the three-dimensional delta "function" has indeed dimension $L^{-3}$.