Why Bare Electron has Infinite Charge and Mass

This concept of bare electron and virtual particles reaching
infinite values is really bizarre to the max so I'll quote the
sources from an interesting particle physics book:

"When the virtual photon fluctuates into the virtual electron
positron pair, its energy must be shared, part going to the
electron and part to the positron. But there are a large
number - in fact, an infinite number - of different ways
that energy can divide itself between the positron and
the electron. The energy of one of the two can be pretty
much anything, as long as the other has the correct
amount of energy to compensate and to make the
total add up to the virtual photon's energy. When
all these different possibilities are taken into account,
the interaction probability of figure 4.8c, instead of
being 10,000 smaller than that of figure 4.4c, is
larger - much larger. In fact, because there's an
infinite number of ways for the process to take place,
the calculated interaction probability is infinite.

This is an unacceptable state of affairs. When this
more complicated diagram is included (which it
must - recall Feynmann's rule that anything that
can happen must happen), we find that our quantum
field theory cannot predict the relative probability of
electrons scattering off each other into the various
directions that you might mount a detector because
the probability of scattering into any given direction
is predicted to be infinite. Since the most that
a probability can be is one, something is wrong.

So, after all its great insights and advancements,
some of them strikingly confirmed by experiments,
are we to conclude that quantum field theory is
useless, having been pecked to death by these
ephemereal matter-antimattter fluctuations?

Luckily, no. But it took some serious soul searching
by Feynman and others to recognize the source
of the problem and to dream up a fix for it. The
approach that we eventually successful in overcoming
this hurdle is known as renormalization"

(back to me)
Well.. in renormalization, the charge of the bare
electron is made infinite to make everything come
out right. Now listen to this stuff by Steven Weinberg's
"Dream of a Final Theory" about the bare electron
infinite mass:

"[The total energy of the atom] depends on the bare
mass and bare charge of the electron, the mass and
charge that appear in the equations of the theory before
we start worrying about photon emissions and reabsorptions.
But free electrons as well as electrons in atoms are
always emitting and reabsorbing photons that affect
the electron's mass and electric charge, and so the
bare mass and charge are not the same as the
measured electron mass and charge that are listed
in tables of elementary particles. In fact, in order
to account for the observed values (which of course
are finite) of the mass and charge of the electron,
the bare mass and charge must themselves be
infinite. The total energy of the atom is thus the
sum of two terms, both infinite: the bare energy
that is infinite because it depends on the infinite
bare mass and charge, and the energy shift ... that
is infinite because it receives contributions from
virtual photons of unlimited energy."

(back to me)
So you see, a bare electron charge and mass is
infinite.. something not even Wilson Renormalization
Group can get rid of. So I just wonder what is
the source of the bare electron infinite charge
and mass. What do you think? Initially I think it's
something akin to black hole singurality but
in reality it may involve more exotic physics.