Do quasars exist or are they a relic of a past aeon?
I would say they are mostly a relic of a past aeon mostly, but the Universe is a big and rather random thing so it does play tricks and they are also here with us (though different naming) and they could be part of the distant future as well.
Most of the quasars are around 10 billion light years away.
The closest (and by definition youngest) quasar that we are aware of is around 2 billion light years away, which is rather young but no closer and younger counterpart was seen so far.
Most galaxies close and far, have been proven to have a supermassive black hole at their center but you need a lot of material to fall into it in order to have an active galactic nucleus.
Only a small fraction of galaxies have sufficient matter in the right kind of orbit at their center to become active and power radiation in such a way to be seen as quasars.
This also explains why quasars were more common in the early universe, as this energy production ends when the supermassive black hole consumes all of the gas and dust near it. This means that it is possible that most galaxies, including the Milky Way, have gone through an active stage, appearing as a quasar or some other class of active galaxy that depended on the black hole mass and the accretion rate, and are now quiescent because they lack a supply of matter to feed into their central black holes to generate radiation.
Now, on to the trick that the Universe can play!
Quasars may also be ignited or re-ignited when normal galaxies merge and the black hole is infused with a fresh source of matter. In fact, it has been suggested that a quasar could form when the Andromeda galaxy will collide with our own Milky Way in 3–5 billion years from now.
Also, since 1980, the consensus in the community was that blazars and quasars are mostly the same thing but viewed at a different angle, blazars having the relativistic jets oriented close to the line of sight with the observer. The closest one is around 300 million light years away, so it is as fresh as it gets!
So, past, present and future, depending on who is watching, they are still somewhat around!