Part One: Star Formation:
It is usual for many stars to form at once out of a dense molecular cloud in space, forming a cluster of stars which gradually dissipates as stars are pulled out of the cluster by the gravity of passing stars.
The apparent magnitude of a star is the brightness which a star appears to have. Depending on human eyesight and viewing conditions, people can see stars with an apparent magnitude down to about 6 or 7. I note that the lower the magnitude, the greater the brightness.
The apparent magnitude of a star depends on its actual luminosity or brightness - its absolute magnitude - and its distance from Earth.
The absolute magnitude is defined as being equal to the apparent magnitude a star with that luminosity would have at a distance of 10 parsecs (32.6156 light years).
Stars which are closer to the Sun than 10 parsecs have apparent magnitudes lower than their absolute magnitudes (thus appearing brighter) and stars which are farther than 10 parsecs from the Sun have apaprent magnitudes which are higher than their absolute magnitudes (and thus appear dimmer).
Amoung main sequence stars the luminosity gets lower along the sequence of types: O, B, A, F, G, K, & M. And the luminosity decreases within a spectral class from the subcass numbered 0 to the sublass numbered 9.
61 Cygni A has an absolute magnitude of 7.506, and a spectral type K5V. So at a distance of 10 parsecs it would too dim to be seen by any except someone with really great vision with really good seeing conditions.
So about half of the spectral calss K stars are too dim to be seen by the unaided eye at a distance of "only" 10 parsecs (which is a vast distance, but very close by galactic standards). And all of the main sequence class M stars are even dimmer. The rare giant and supergiant class M stars are much mighter and seen at much greater distances, but they are old stars past the main sequence stage, not newly formed stars which just lit up.
Class M stars are by far the most common. About 76% of the main-sequence stars in the solar neighborhood are class M stars.[e][f] However, class M main-sequence stars (red dwarfs) have such low luminosities that none are bright enough to be seen with the unaided eye, unless under exceptional conditions. The brightest-known M class main-sequence star is Lacaille 8760, class M0V, with magnitude 6.7 (the limiting magnitude for typical naked-eye visibility under good conditions is typically quoted as 6.5), and it is extremely unlikely that any brighter examples will be found.
So most new stars which form will be too dim to be seen from EArth, unless they form really close to Earth.
The nearest star forming regions include the Orion Nebula at about 412 parsecs or about 1,340 light years, The Taurus Molecular Cloud at about 140 parseces or 430 light years, The Rho Ophiuchi Complex at about 130 parsecsor about 420 light years; and the Corona Australis Molecular Cloud at 130 parsecs or 430 light years.
At the distance of the Orion Nebula, a star would have to have a absolute magnitude of 0 to have an aparent magnitude of 7, and an absolute magnitude of - 1 to have an apparent magnitude of 6. So the vast majority of the stars formed in the Orion Nebula would far too dim to be visible from Earth without telescopes.
At the distance of the Rho Ophiuchi Complex or the Corona Australis Molecular Cloud a star would have to have an absolute magnitude of 1.5 to have an apparent magnitude of 7, and an absolute magnitude of 0.5 to have an apparent magnitude of 6. So the vast majority of the stars formed in those regions would far too dim to be visible from Earth without telescopes.
Suppose that on the average one new star suddenly lit up (ignoring whether stars do suddenly light up) in a star forming region every 1,000 years. Since the vast majority of those stars would be too dim to be seen from Earth, it might take 10,000 years or 100,000 years on the average for one of those new stars to be seen with the unaided eye from Earth by prehistoric, ancient or medieval, people and be added to any star charts they might make.
It is statistically improbable that humans or earlier members of Genus Homo would have noticed such a hypothetical suddenly igniting star and remembered it down the millennia until people started writing about the stars just a few thousand years ago. After a few centuries or millennia they would probably forget that it suddenly appeared and think that it had always been seen by their ancestors.
Part Two: Stars Coming within visual Range.
Stars orbit around the center of mass of the Milky Way Galaxy in incredibly vast orbits.
And because the orbits of even the stars closest enough to each other can not be perfectly identical (because that would make the two stars be in the same place at the same time) two stars will get closer to each other, pass, and then get farther away from each other, much like two planets orbiting a star.
This list shows various close passes between the Sun and other stars in the past few million years and in the next few million years.
The Sun has 1.00 times the luminosity of the Sun, and an absolute magnitude of 4.83.
The magnitude scale is logarithmic, with a difference of 1 magnitude being a difference of 2.512 times brightness, and a difference of 5 magnitudes being a difference of 100 times brightness.
Many of the stars on the list which once approached within 5 light years, or will approach that close within the next few million years, are now too far from Earth to be visible to the naked eye. Binoculars or telescopes are needed to see them.
But some them did, or will, approach close enough to be seen with the naked eye.
For example HD 49995 is now about 439 light eyars from Earth, but was only about 1.7 light years distant about 4,034,000 years ago. So it was only about 1/258 as far from Earth, and would have appeared 258 X 258, or about 66,564 times, as bright as it appears now. Since it now has an apparent magnitude of 8.78, it shoud have had an apparent magnitude of about minus 3.22, brighter than Sius or the planet Jupiter.
So over a period of tens or hundreds of thousands of years, HD 49995 became just barely bright enough to see, got brighter until it was brighter than any other star in the sky, and then got dimmer and dimmer until it was too dim to see.
And that has happened to all the stars which have ever been visible from Earth with the unaided eye, and will happen to all the stars which will ever become visible from Earth with the unaided eye. And I suspect it's incredibly unlikely that has ever happened fast enough to be noticed within historical record.