From Gauss's law of gravity reduced to 2+1 dimensions, one can easily show that the gravitational force follows an inverse law, i.e. $$ \mathbf{F}(\mathbf{r}) =- \frac{G m M}{|\mathbf{r}|}\hat{\mathbf{r}}. $$ Similarly, one can derive that the gravitational potential $V$ at a distance $r$ from a point mass of mass $M$ reads $$ V(r) = G M \log(r). $$
However, the physical interpretation that the gravitational potential can be defined as the work that needs to be done by an external agent to bring a unit mass from infinity to the distance $r$ from a point mass $M$ now fails since $$ V(r) = -\frac{1}{m}\int_{\mathbf{\varphi}} \mathbf{F} \cdot \mathrm{d}\mathbf{s} = \int_{\infty}^{r} \frac{GM}{r'} \mathrm{d}r' = GM \left[ \log r' \right]_{r' = \infty}^{r} = \infty. $$ Is there a way to reconcile this? Or is it fundamentally wrong to try to reduce Newton's gravity (or similarly, Gauss's law of electrostatics) to 2+1 dimensions?
This question is different from What is the 2+1D gravity potential? where it is explained why the gravitational force follows an inverse law in 2+1D (instead of inverse square law known from 3+1D) but the work done by the gravitational force field "from infinity" is not discussed.
