The FDTD, belongs to a kind of explicit method, is efficient if the Courant-Friedrichs-Lewy condition is satisfied, which is difficult to do in static analysis.
The FDTD would only be effective for high-frequency electromagnetism.
High-frequency electromagnetism and low-frequency electromagnetism (for which the implicit method is efficient) should be considered as two separate fields that differ greatly in both theory and numerical calculations.
EDIT on Sat Mar 11 16:27:08 JST 2023.
i wanted to compensate my description.
The short conclusion is, "Calculating low-frequency problems with the FDTD method would simply break down due to the extremely large number of computational steps.".
Let's focus on 2 problems illustrated in the Figure.
On the left of the Figure is a schematic diagram of an antenna problem radiating radio waves.
Suppose the drive frequency is, for example, 1MHz and the horizontal length of the antenna is assumed to be about half the wavelength.
On the right of the Figure is a schematic diagram of a static capacitor problem.
And suppose that the metal rod configuration is unchanged from the left case.
If the space configurations are the same between 2 cases,
we want to see if we can get the electric field for the right case using FDTD or not.
The left problem can be solved by the FDTD method.
In this case, the CFL condition, has the following form,
\begin{equation}
K=\frac{c\Delta t}{\Delta x}\leqq K_{\text{max}} \;(\lesssim 1.0),
\end{equation}
which should be satisfied,
where $c$ is the speed of light, $\Delta t$ is the time step and $\Delta x$ is the length interval.
The larger $\Delta t$, the shorter the computation time, but this CFL inequality gives an upper bound on $\Delta t$. It is not permissible to increase $\Delta t$ in order to economize on computation time.
Suppose we have solved the left Figure case. In this case, the number of computation steps is calculated as
\begin{equation}
N=\frac{T}{\Delta t} =\frac{cT}{c\Delta t}\geqq \frac{cT}{K_{\text{max}}\Delta x}
\left(=\frac{1}{K_{\text{max}}}\frac{\lambda}{\Delta x}\right)
\end{equation}
where $N$ is the number of calculation steps corresponding to one A.C. period, and $\lambda$ is the wave length.
And inequality $\geqq$ comes from the CFL condition.
Let us assume the following expression for the left Figure case for later use:
\begin{equation}
\frac{N}{T}\simeq\frac{c}{K_{\text{max}}\Delta x}
\end{equation}
Next, consider solving the case on the right model in the Figure,
with FDTD using the same mesh partitioning model;
choosing the same partitiong model is an assumption.
The symbols ${}'$ stands for the right Figure case.
As an example, say, we choose the frequency of 1 micro Hz,
we expect to approach a static answer.
The CFL condtion for the right Figure case is
\begin{equation}
\frac{c\Delta t'}{\Delta x'}\leqq K_{\text{max}},
\end{equation}
and let us add 2 assumptions as follows
\begin{equation}
\Delta x'= \Delta x
\end{equation}
\begin{equation}
T' = 10^{12}T
\end{equation}
Using these inequality and expressions, we get
\begin{equation}
N' = \frac{T'}{\Delta t'}
\geqq T'\left(\frac{c}{K_{\text{max}}\Delta x'} \right)
= T'\left(\frac{c}{K_{\text{max}}\Delta x} \right)
\simeq T'\frac{N}{T} = 10^{12}N
\end{equation}
Since this number $N'$ is extremely large, it is very hard to calculate using real world computers, and second, FDTD is based on an explicit method, error accumulation is not small,
making it virtually impossible to solve low-frequency problems unless the object length is very very long.
Best Answer
The main problems in solving directly the Gauss's law are:
In order to have a well defined problem, that satisfies the irrotational condition, you exploit it to write the electric field as the gradient of a scalar field, the potential field $\Phi(\mathbf{r})$, defined as $\nabla \Phi(\mathbf{r}) = \mathbf{e}(\mathbf{r})$, and you substitute this latter expression in the Gauss's law for the electric field to get the Poisson's equation
$\nabla^2 \Phi(\mathbf{r}) = \dfrac{\rho(\mathbf{r})}{\varepsilon_0}$.
This is a PDE define in a domain, and you need boundary conditions on the boundary of the domain to get a well-defined problem. The choice of the boundary conditions really depends on the problem you're studying:
if your domain is the whole 3D space, and you're only interested to a limited regions that includes all the electrical charges that influence your system, you will probably use the condition of vanishing field $\Phi(\mathbf{r}) \rightarrow 0$ at infinity; from the analytical or numerical point of view, you hardly approach this problem with finite difference method, but you could easily treat it with singularity methods, boundary element methods, or other methods relying on a integral formulation of the problem using the Green's function method;
if your domain is bounded, you can approach the problem with classical grid based numerical methods (like finite differences, finite elements, finite volumes, discontinuous Galerkin methods, ...) but you need to model a representative boundary condition of the electric field and translate it to the potential field. As an example,
on a surface ${S_N}$ where you know the normal component of the electric field, you can translate this condition into a boundary condition on the directional derivative of the potential field in the direction that is normal to the boundary (Neumann or natural boundary condition), as
$e_n(\mathbf{r}_{S_N}) = \mathbf{\hat{n}}(\mathbf{r}_{S_N}) \cdot \mathbf{e}(\mathbf{r}_{S_N}) = \mathbf{\hat{n}}(\mathbf{r}_{S_N}) \cdot \nabla \Phi(\mathbf{r}_{S_N}) = \dfrac{\partial \Phi}{\partial n}(\mathbf{r}_{S_N})$.
on a surface ${S_D}$ where you know the potential w.r.t. to an irrelevant offset value, you can prescribe the value of the potential (Dirichlet or essential boundary condition)
$\Phi(\mathbf{r}_{S_D}) = \Phi_s(\mathbf{r}_{S_D})$,
and if the surface is equipotential, so that no current occurs on it, $\Phi(\mathbf{r}_{S_D}) = \Phi_{S_D}$, with $\Phi_{S_D}$ constant on the surface ${S_D}$
Take a look at this reference for a comparison between finite element and boundary element method: https://hrcak.srce.hr/file/265787