Work done in assembling a point charge is infinite Griffiths says

The energy of a point charge is infinite
$$W=\frac{\epsilon_{0}}{2\left(4 \pi \epsilon_{0}\right)^{2}} \int_0^\infty\left(\frac{q^{2}}{r^{4}}\right)\left(r^{2} \sin \theta\; \mathrm{d}r\;\mathrm{d}\theta \;\mathrm{d}\phi\right)=\frac{q^{2}}{8 \pi \epsilon_{0}} \int_{0}^{\infty} \frac{1}{r^{2}} d r=\infty$$

And then he says that this infinity is an embarrassing failure of electromagnetic theory. But what's so embarrassing about it?
It's obvious that if you pack same charges close and closer together we will have to work harder and harder. And in the case of an almost zero volume of a point charge the work we would have to do will go to infinity; hence the above result.  So what's the embarrassment?
 A: Griffiths was referring to "the problem of infinities". See the Wikipedia article on the history of quantum field theory which has a section on it here.
People often say this issue stemmed from classical electrodynamics, but I'm not sure it does. As far as I can tell it stems from Yakov Frenkel's 1925 paper The electrodynamics of rotating electrons. Frenkel said “The electron will thus be treated simply as a point”. This was popularized by Heisenberg and Pauli who saw people like de Broglie and Schrödinger as the competition. See what de Broglie said in his 1923 letter to Nature on waves and quanta: “the wave is tuned with the length of the closed path”. See what Schrödinger said on 26 of quantization as a problem of proper values, part II: “let us think of a wave group of the nature described above, which in some way gets into a small closed ‘path’”.
The point-particle electron was popularized despite the evidence for the wave nature of matter provided by the 1923-1927 Davisson-Germer experiment and the contemporaneous Thomson and Reid diffraction experiment. Despite the fact that physicists like Oppenheimer said it was wrong. See his 1930 note on the theory of the interaction of field and matter where he said “the theory, is, however, wrong, since it gives a displacement of the spectral lines… which is in general infinite”.
People didn't say this sort of thing about classical electromagnetism which dealt with waves and fields and was largely developed before the electron was discovered. Even after the electron was discovered, people like Gustav Mie weren't saying it was a point particle. In his 1913 Foundations of a theory of matter, he talked about four-potential embodying the state of the ether and knot singularities in the field. I like the sound of that myself, it reminds me of TQFT.
A: In the annihilation of an electron, energy is bounded by 511keVs so the theory has a problem that opposes the experimental results. Where I think that the classical theory is not an embarrassment instead where it fails is an indicator of where the theory fails about predicting charges as point particles with infinite energy. So, the theory fails to predict certain facts but failure was so apparent that even it was a clue to a new area to explore.
A: “Embarrassing” is an opinion, and Griffiths generally has very well-respected opinions. However, as opinions are not facts it is not necessary for everyone to share them.
To me, it is not an embarrassment but rather an early hint that the electron is not a classical point particle. A classical point particle has infinite energy, an electron does not have infinite energy, therefore an electron is not a classical point particle.
There are also several other classical EM paradoxes that are based on the idea of a classical point particle. To me, all of these should be taken not as a failure of classical EM but as a failure of the classical point particle concept.
A: In addition to the experimental fact from Butane, infinite self-energy becomes problematic when considering relativity- where absolute energies matter, and not just relative ones.
Note that the energy of a sphere of finite radius around the electron would still be infinite, as $\int_0^1 1/r^2 dr = \infty$. An electron with infinite self-energy bound to a finite region would have infinite mass, with an infinite Schwarzschild radius.
A: 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.
