• if the charged particle that is experiencing EM force is a point, the term $\mathbf j \cdot \mathbf e$ is too singular and its meaning cannot be consistently defined; total electric field diverges at the particle. So the assumption about $\mathbf j\cdot \mathbf e$ is invalid and Poynting's theorem while true, is useless. We can derive another similar theorem that can be interpreted as local conservation of energy, if we base it on some regular expression for EM work. There is the Frenkel theory which defines its own energy density and energy flux density expressions based on the idea that point particles experience only external field, and their own field does not affect them. This leads to expressions for EM energy and EM energy flux density that are different from the Poynting ones.

• if the charged particle is extended in space with a finite charge density everywhere, the term $\mathbf j \cdot \mathbf e$ is fine and Poynting's vector formula is fine, but $\mathbf e$ is usually unknown to us, because near and inside the particles, it is affected by the particle's own field in a non-trivial way:

When electric field accelerates a charged particle, the particle generates magnetic field component proportional to the velocity everywhere near and inside the particle, and total magnetic field no longer vanishes where the particle is. This way, the particle's own magnetic field due to its motion makes it possible for vector of EM energy flux to not vanish, and for electric energy to be transformed into kinetic energy of the particle. This is the case irrespective of whether charged particle is a point or extended charge distribution with finite density.

• if the charged particle that is experiencing EM force is a point, the term $\mathbf j \cdot \mathbf e$ is too singular and its meaning cannot be consistently defined; total electric field diverges at the particle. So the assumption about $\mathbf j\cdot \mathbf e$ is invalid and Poynting's theorem while true, is useless. We can derive another similar theorem that can be interpreted as local conservation of energy, if we base it on some regular expression for EM work. There is the Frenkel theory which defines its own energy density and energy flux density expressions based on the idea that point particles experience only external field, and their own field does not affect them. This leads to expressions for EM energy and EM energy flux density that are different from the Poynting ones.

• if the charged particle is extended in space with a finite charge density everywhere, the term $\mathbf j \cdot \mathbf e$ is fine and formula for the Poynting vector is fine and can be interpreted in terms of local conservation of energy, but $\mathbf e$ is usually unknown to us, because near and inside the particles, it is affected by the particle's own field in a non-trivial way:

When electric field accelerates a charged particle, the particle generates magnetic field component proportional to the velocity everywhere near and inside the particle, and total magnetic field no longer vanishes where the particle is. This way, the particle's own magnetic field due to its motion makes it possible for vector of EM energy flux to not vanish, and for electric energy to be transformed into kinetic energy of the particle. This is the case irrespective of whether the charged particle is a point or extended charge distribution with finite density.

Such interpretation of the Poynting theorem equation is incorrect or correct but unfeasible in microscopic EM, where total fields are $\mathbf e, \mathbf b$, for the following reasons:

• if the charged particle that is experiencing EM force is a point, the term $\mathbf j \cdot \mathbf e$ is too singular and its meaning cannot be consistently defined; total electric field diverges at the particle. So the assumption about $\mathbf j\cdot \mathbf e$ is invalid and Poynting's theorem while true, is useless. We can derive another similar theorem that can be interpreted as local conservation of energy, if we base it on some regular expression for EM work. There is the Frenkel theory which defines its own energy density and energy flux density expressions based on the idea that point particles experience only external field, and their own field does not affect them. This leads to expressions for EM energy and EM energy flux density that are different from the Poynting ones.

• if the charged particles are extended charge distributions, the term $\mathbf j \cdot \mathbf e$ is fine and Poynting's vector formula is fine, but $\mathbf e$ is usually unknown to us, because near and inside the particles, it is affected by the particle's own field in a non-trivial way:

People sometimes assume (as you did) they can simply put into the Poynting formula the macroscopic field $\mathbf E$ or total external microscopic field $\mathbf e_{external}$ and obtain the correct energy flux density. But that is not correct. The electric field due to extended charged particle itself is contributing to the total electric field acting on the particle; particle's own field acts and can do work on the very same particle. This energy exchange is not captured by Poynting's vector based on macroscopic field $\mathbf E,\mathbf B$, or based on external microscopic fields $\mathbf e_{external},\mathbf b_{external}$.

In your example, Poynting's vector vanishes at a certain point of space, while electric field does not. But that is possible only if there is no moving charged particle there. If there is some moving charged particle or group of moving charged particles there, these will affect the total microscopic field so that magnetic field does not vanish. Poynting's vector based on microscopic fields won't vanish there anymore.

When electric field accelerates a charged particle with finite charge density, the particle generates magnetic field component proportional to the velocity, and total magnetic field no longer vanishes where the particle is. This way, the particle's own magnetic field makes it possible for electric energy to be transferred into kinetic energy of the particle, in agreement with either Poynting's or Frenkel's expressions for EM energy and EM energy flux density.

Such interpretation of the Poynting theorem equation is either incorrect, or correct but more complicated in the microscopic EM theory, where total fields are $\mathbf e, \mathbf b$, for the following reasons:

• if the charged particle that is experiencing EM force is a point, the term $\mathbf j \cdot \mathbf e$ is too singular and its meaning cannot be consistently defined; total electric field diverges at the particle. So the assumption about $\mathbf j\cdot \mathbf e$ is invalid and Poynting's theorem while true, is useless. We can derive another similar theorem that can be interpreted as local conservation of energy, if we base it on some regular expression for EM work. There is the Frenkel theory which defines its own energy density and energy flux density expressions based on the idea that point particles experience only external field, and their own field does not affect them. This leads to expressions for EM energy and EM energy flux density that are different from the Poynting ones.

• if the charged particle is extended in space with a finite charge density everywhere, the term $\mathbf j \cdot \mathbf e$ is fine and Poynting's vector formula is fine, but $\mathbf e$ is usually unknown to us, because near and inside the particles, it is affected by the particle's own field in a non-trivial way:

People sometimes assume (as you did) they can simply put into the Poynting formula the macroscopic field $\mathbf E$ or total external microscopic field $\mathbf e_{external}$ and obtain the correct microscopic energy flux density. But that is not correct. The electric field due to extended charged particle itself is contributing to the total electric field acting on the particle; particle's own field acts and can do work on the very same particle. This energy exchange is not captured by Poynting's vector based on macroscopic field $\mathbf E,\mathbf B$, or based on external microscopic fields $\mathbf e_{external},\mathbf b_{external}$.

In your example, macroscopic Poynting's vector vanishes at a certain point of space, while macroscopic electric field does not. But that is possible in microscopic theory too only if there is no moving charged particle there. If there is some moving charged particle or group of moving charged particles there, these will affect the total microscopic field so that magnetic field does not vanish. Poynting's vector based on microscopic fields won't vanish there anymore and EM energy flux into or from the charged particle can be non-zero.

When electric field accelerates a charged particle, the particle generates magnetic field component proportional to the velocity everywhere near and inside the particle, and total magnetic field no longer vanishes where the particle is. This way, the particle's own magnetic field due to its motion makes it possible for vector of EM energy flux to not vanish, and for electric energy to be transformed into kinetic energy of the particle. This is the case irrespective of whether charged particle is a point or extended charge distribution with finite density.