Electronic – Charged capacitor doesn’t allow impulses to pass

That article is crap, and you should probably forget you ever saw it. Besides the fact that it's totally wrong, as you've discovered, it's full of half-truths that cultivate an incorrect understanding of how electrical phenomena actually work.

The plate on the capacitor that attaches to the negative terminal of the battery accepts electrons that the battery is producing.

wrong. Batteries don't produce electrons. In fact, nothing in your circuit produces electrons. As far as physicists have been able to demonstrate, charge is never created nor destroyed. A physicist can tell you how to make an electron by assembling more fundamental particles, but unless your circuit includes a particle accelerator or operates inside a star, there won't be any relevant electron creation or destruction in your circuit. Batteries pump electrons. They don't produce them.

In the first few paragraphs, that article uses the word charge in several senses. An experienced engineer can distinguish the senses by context, but the novice is more likely to confuse them. Go read Bill Beaty to immunize yourself against this misconception.

The bulb will get progressively dimmer and finally go out once the capacitor reaches its capacity.

Not exactly true. The bulb will go out once the voltage across the capacitor is equal to the battery voltage. When this happens, there can be no voltage across the bulb, and thus no current, thus no light. I don't know what they mean by capacity in this sense, but it seems to me that they are phrasing it this way to avoid explaining how capacitors actually work. You might define a capacitor's "capacity" any number of ways, but they haven't defined it at all. This kind of half-thinking isn't going to help you understand once you figure out that the explanation is incomplete.

A capacitor's storage potential, or capacitance, is measured in units called farads.

If you take "capacity" and "storage potential" as synonymous, as anyone would naturally do, this is wrong and self-contradictory. If this were true, then we could re-write the previous statement as

The bulb will get progressively dimmer and finally go out once the capacitor reaches its capacitance

...which is totally bogus. Ordinary capacitors don't change capacitance under normal operation conditions. Again, the problem here is they haven't fully defined the underlying concepts. If capacitance is a capacitor's storage potential, then what is it storing?.

A 1-farad capacitor can store one coulomb (coo-lomb) of charge at 1 volt.

Here, they try to define the stuff that the capacitor is storing, but it hardly makes sense. The problem is if you use words like "capacity", this implies there is some sort of "full" or "at capacity". But, there is no concept for an ideal capacitor. If you push 1C of charge through a 1F capacitor, the capacitor will have a voltage of 1V. For 2C, you get 2V, and 1000C gets you 1000V. You can push 1000C through a 0.5F capacitor also, but it will then be at 2000V. An ideal capacitor is never full, and you can push charge through it forever and it will never be "full". That's not a capacity, it's the ratio of charge to voltage, which is evident in the definition of farad (a farad is a coulomb per volt):

$$ F = \frac{C}{V} $$

Real capacitors, incidentally, have a maximum voltage, which if exceeded will damage or destroy the capacitor. So in this sense, a capacitor can be "full" and can have a "capacity". But, this is not how this article is using the word.

I could probably write an entire article on the problems with that article, but hopefully you get the idea. Please wipe your memory of what that article has said, and seek a more sound explanation.

I don't understand how a component that blocks current once it is charged could be useful.

To be precise, the current through a capacitor is zero ('blocked') if the voltage across the capacitor is constant.

It turns out that this is enormously useful for, e.g., coupling AC signals between circuit functional sections without disturbing the DC operating point.

For example, it may be that the output of an amplifier is a constant 5V plus some time varying (AC) component.

The output is to be connected to, e.g., a speaker that cannot tolerate a constant 5V across (as it would 'burn out').

If an appropriately sized capacitor is connected in series between the amplifier output and speaker, there will be no current through the capacitor and speaker due to the constant 5V at the amplifier output.

However, the time varying component will cause a time varying current through the speaker and, thus, some type of sound will be produced.

Such a capacitor application is called capacitive coupling.

Sometimes, the opposite is required and this called capacitive decoupling.

Rather than connect a capacitor in series, a capacitor is connected in parallel with, e.g., a power supply. In this case, the time varying components are undesired and are shunted by the capacitor 'around' the load.

A related and overlapping application is capacitive bypass.

There is much more to be said about capacitor applications which are manifold as a perusal of the Wikipedia article "Capacitor" will reveal.