quantum field theory - Divergent self-energy of point charges in Classical Electrodynamics

Assuming the electron to be a classical point particle, if one calculates the self-energy one finds

$$U=-\frac{e^2}{8\pi\epsilon_0r}$$ which diverges as $r\rightarrow 0$. Therefore, the measured mass of the electron should be $$m_{0e}+U/c^2=m_e.$$

Now, $m_{0e}$ is the mass of the electron in absence of its electric field (which is therefore unobservable because its electric field cannot be switched off) and $m_e$ is the measured electron mass.

Why is it a problem that in classical electrodynamics, the self-energy of a point electron diverge? The divergence in $U$ may be absorbed in $m_{0e}$ which is unobservable as often done in renormalization.

This is presented as a problem in textbooks because often one equates $m_ec^2=U$, and forget about $m_{e0}$. But I don't see a reason why $m_{e0}$ should be neglected. See page 28 here.