Of course, it's an embarrassment. When the charge is not point-like the energy will be non-infinitely. Just like the energy of a point-like charge. Of course, when you bring two electrons together from infinity to zero (as in the cited formula in the question), they'll have each an infinite energy, but a single electron will not.
A classical point particle does not have infinite potential energy, and thus no infinite mass, as experiments show. This is because it is not formed by putting together sub-charges. So these charges can't
When the equation amounts to starting from an infinite charge distribution with a total charge of -1, a big physical mistake is made. This is simply not how point-charges are formed (or non-point-particles).
Integrating over the electron's static force (in 3d), gives indeed an infinite value. This isn't the energy of the electron though. It's the energy you get when pushing charge towards the electron.
So it's indeed an embarrassment.
So it's reasonable to ask, in the classical context, how come the charge doesn't explode?
From the previous answer we can conclude that in the case of infinite energy, we all would be living in multiple black holes, which obviously is not the case.
As an extra
In QFT, the electron's charge is quantized but is still is seen as point particles. It follows from the math. And also (from this lecture):
In QED, the bare charge of an electron is actually infinite!!! Note: due to the field-energy near an infinite charge, the bare mass of the electron (E=mc2) is also infinite, but the effective mass is brought back into line by the virtual pairs again !!
That is, just as in classical EM. This problem is "solved" by renormalization, of which I'm very suspicious, even in its present form (the Wilsonian approach); renormalization stays renormalization. I'm suspicious because renormalization uses other electric charges to make the charge of the electron finite.
There is a theory in which the electron is composed out of three particles of charge $-\frac 1 3$. This is the Rishon model. Three anti-T-rishons form the electron. These three electric charges are held together by a force stronger than the electric force that repels them. The three charges obviously don't have infinite potential energy.
So, in this model, the problem is: do T-rishons have infinite potential energy? Again the answer is no. These particles, just as in the case when the electrons are considered a point-particle, are not formed by putting together sub-charges.
I don't think elementary particles are point-like. I'm not talking about string theory, in which charge is somehow connected with vibrations of whatever kind of brane or manifold. I have my own ideas about the non-point-like structure of elementary particles but I won't bother you with them.