**It is not true that $\dot\rho(\phi_1,\phi_1)=0$: the object $\rho(\phi_1,\phi_1)$ does not exist.**

It doesn't make sense to talk about $\rho$ for $\phi_1=\phi_2$. The object $\rho$ is a distribution (as a function of $\phi$, not $x$) so it is meaningless to "evaluate" it on a given function.

Indeed,
$$
\langle\phi_1|\phi_2\rangle=\delta(\phi_1-\phi_2)
$$
where $\delta$ is the Dirac delta in the space of functionals (i.e., with respect to functional integration, not regular integration). See [this PSE post](https://physics.stackexchange.com/a/383473/84967) for some (formal) applications of this formula.

The density matrix $\rho(\phi_1,\phi_2)$ only makes sense under the integral sign:
$$
\int_{\mathcal F^2}\rho(\phi_1,\phi_2)F[\phi_1,\phi_2]\,\mathrm d\phi_1\mathrm d\phi_1
$$
where $\mathcal F$ denotes some reasonable function space and $F$ is a healthy functional on it.