If you divide a small number by a much smaller one, you end up with something resembling a infinity, eventually.
There are two kinds of singularities related to black holes.
The first kind are coordinate singularities, for example at the event horizon in the Schwarzschild metric. Those are mathematical artifacts of the chosen coordinates and we can get rid of them by using different coordinates for calculations.
The second kind are the ones at the center, where, for example, the curvature blows up to infinity. For those we do not know how to get rid of them. Still very few physicist will consider them any more real than the first kind, they are most likely an artifact of relativity not accounting for quantum physics or something like that.
When it comes to naked singularities, we are concerned with the second kind of singularity, those at the center. They are usually hidden behind the event horizon because - in case of a non-rotating uncharged black hole - the Schwarzschild radius is positive and nothing from inside the Schwarzschild radius, where the event horizon is located, can reach the outside.
For rotating or charged black holes the event horizon can however disappear, I think the Schwarzschild radius or its equivalent becomes complex. No event horizon, nothing hiding the singularity, i.e. a naked singularity.
So the uncharged non-rotating electron has a mass and I would guess also a singularity in the gravitational field but its Schwarzschild radius is positive hiding this singularity behind the event horizon. That is, if you wanted to get close enough to the electron to have a look at the gravitational singularity, you would first have to pass through the event horizon of the electron and then you could unfortunately no longer tell us what you saw. Also the "electron" would now be 70 kg, I guess.
If I've plugged in the right figures, the Schwarzschild radius would be something like 22 orders of magnitude smaller than the Planck length. (edit - ah the result of this is in the wiki article)
The problem is that the maths of general relativity assumes spacetime is differentiable everywhere, while also predicting non-differentiable singularities from benign starting conditions. This is still a problem no matter the Schwarzschild radius, because the unwanted infinity is still there.
It's also not a problem, because everyone already knows by this point that neither GR nor QM are complete, and stuff on this scale is exactly the sort of thing where a future replacement for both of them takes over.
"Problem" or "not" is just "do you work on finding the next theory, or do you build stuff at scales where we can pretend both (or either one plus Newton) are the complete truth?"
That the Schwarzschild radius is less than the Planck length might mean something or not, all the Planck quantities are just scales derived from fundamental constants. At those scales different effects from our current theories would become comparable if our theories would hold at those scales, but there is currently not really any known significance to them.
It is about the limitations of describing positions, not an implication that spacetime is discrete.
We don't know if spacetime is discrete, but we also have no evidence to suggest it is right now.
If you consider at what energy a photon (E = hc/λ) will be able to resolve, i.e. have a wavelength equal to, the Schwarzschild radius of that energy (r = 2GM/c² with E = mc²), then you get the Planck length up to a factor of root two [1], implying that measuring a Planck distance requires a photon so energetic that it turns that region into a black hole. If, very big if, the known laws still apply at those scales.
[1] To get the Planck length, the wavelength must be doubled, i.e. be equal to the diameter instead of the Schwarzschild radius.
r = 2GM/c² → m = rc²/2G → E = rc⁴/2G (with E = mc²)
hc/λ = rc⁴/2G → hc/2r = rc⁴/2G (with λ = 2r) → r² = hG/c³ → lₚ = r = √hG/c³I don't see how this could work if an electron would be a black hole.
It doesn't, because the "Planck length" and all other "Planck" units ARE NOT fundamental, universal limits. They are merely a differently defined unit that makes some of the math simpler. It's just a really small yard stick, they don't mean anything.