So from this I can only come to the conclusion that supernovae did in
fact happen in the Milky Way in the last two centuries, but that we
didn't see any of them.
But now I'm confused. I read about supernovae that have happened in
galaxies billions of light-years from here that lit up as the
brightest source of light in the sky for days. Surely we would be able
to see all of the supernovae that happen in our Milky Way
One problem with estimating how many supernovae have happened in the Milky Way in the last two centuries, is that the remnant from a supernova is far dimmer than the nova itself and far harder to find. Several supernovae could have happened in the Milky Way in the last 2 centuries and remain undiscovered — and just for clarity, we define the date of a supernova by the date the light from the explosion reaches the Earth, so saying saying a nova "happened" in the last 200 years refers to the date the light from the event reached Earth, not the actual date of the event, which you probably already know, but just to clarify.
So for argument's sake, let's say that a supernova's light reached the Earth about 50 years ago, but it took place on the far side of the galaxy. To find that, we'd have to look for a nebula on the far side of the galaxy and that's a hard thing to see. Similar to looking for the theoretical Planet Nine, finding old supernova remnants on the other side of the Milky Way takes a lot of looking. Even with modern telescopes, it's still a needle in a haystack, and especially if the view is blocked by dust like much of the Milky Way is.
A Milky Way supernova remnant was discovered in 1985, supernova remnant G1.9+0.3. It was thought to have "happened" around 1868, though it went unobserved and probably wasn't visible to the naked eye at the time. There's probably been several others more recent than that one. G1.9+0.3 would have been visible if not for interstellar dust. From article above:
It was a type Ia supernova believed to have exploded about 25,000
years ago, and the signal began reaching Earth around 1868. The light
from the supernova would have been visible to 19th century
astronomers, had it not been obscured by the dense gas and dust of the
I'm not sure the 4.6 supernovae per century from your article is accurate. It might be, but the number I'm used to hearing is about one per century. But regardless of which number is actually correct, it still doesn't imply high mathematical improbability because many Milky Way novae would have gone unnoticed if they were far enough away. In short, I agree with what you said here:
I can only come to the conclusion that supernovae did in fact happen in the Milky Way in the last two centuries, but that we didn't see any of them.
Here's a related article on Milky Way visibility.
Observation of a distant galaxy nova is possible if we have telescopes looking in that direction. A nova is much more detectable at the time it goes nova. Much less so, years later.
Today, however, with neutrino detection in 7 locations around the globe, I think it's virtually impossible that we'd miss a supernova in the Milky Way, so we have a good chance of seeing one in our lifetime and it's likely that none have occurred since somewhere around 1980. A supernova was detected in the Andromeda galaxy in 1987 by that method, and our neutrino detection has improved since then to give us early warning and pinpoint location.
As far as visibility, size matters, but what matters more is how close and how much dust is in the way. Most of the recorded supernovae were quite bright and fairly noticeable to someone who was familiar with the stars in the sky (list of known supernovae).
Eight Milky Way novae have been recorded by history and observed by the naked eye in the last 2,000 years, well below the number that should have happened in that time. Five of those eight had brightness greater than $-3$, which is brighter than Jupiter and would have been immediately noticed by anyone familiar with star charting. Two others had magnitudes around zero, which is still brighter than most stars. SN386 was less bright, but is still easily visible and recorded by Chinese astronomers. And finally, Cassiopeia A was quite dim, but it was still observable. Only about 10% (ballpark estimate) of the Milky Way is close enough and unobstructed enough to provide supernovae that would visibly get noticed. Most would have gone unnoticed until recently.
Hope that's not too wordy, I can try to clean up if needed.