Rather than saying no actual infinity exists in our universe, it would be more precise to say that we can only see a finite portion of the universe, so we can't tell if the full universe beyond our sight contains infinities or not.
We see a piece of the universe, but we can't see the edges. It extends out of sight in all directions. So we can only speculate as to what might be out there, beyond the circle of firelight. We can count atoms going 1, 2, 3, 4, ... and so on. What happens at the edge? Is there some specific number of atoms to which it is impossible to add another? How/why? Is there some specific number such that when you add another atom they all disappear, and you start again at zero? Or something stranger? Or might it simply keep going without end? We have no data, how do we decide?
Infinity is often the simplest option. Having to specify a limit, and then specify the behaviour at the limit that stops you going any further, introduces arbitrary complexities into our model of the world. We have no evidence on which to base such speculations or to test their truth. So we pick the one that is simplest to work with, that is sufficiently accurate in its predictions for our purposes, but remain open to other possibilities.
On your question "whether the possibility that "infinity" simply comes as a byproduct of our brains, and not as a feature of reality has been given any thought in philosophical literature" - the literature on infinity has tended to operate the other way round. Since the very beginning it has repeatedly been denounced as illusory and inconsistent, a mental construct so rife with paradoxes and contradictions that it is barely credible even as an idealised mental model of the world, or Platonic ideal, let alone as reality. It has taken centuries of effort by mathematicians to rehabilitate it - to find ways to understand it as a respectable and internally consistent 'number' that we can do maths with same as any other number. Indeed, it has only relatively recently been found that the ordinary numbers appear to be 'broken' or incomplete without it. (For example, projective geometry adds 'points at infinity' to Euclidean geometry, resulting in a much more elegant and symmetric theory.) We have only just recently got to the point where we can take seriously the possibility that reality might actually be infinite!
And it's worth pointing out that the rules brains and Turing machines operate by are themselves the rules of the universe. The universe itself makes computation possible, and makes concepts like infinity conceivable. Why, then, should it not make infinity possible? We have found out so much about the universe by examining a tiny piece of it, figuring out the rules it follows, and then extending the logical consequences of those rules to the rest. (E.g. it's how we found out about the Big Bang, and that the universe is only 10-15 billion years old.) It's true that the only place we can observe infinities is in brains and Turing machines, but it is a bold claim to say this is merely an 'accident', that tells us nothing about the way the universe works. We can't see infinity, but we can see the rules in the part we can see, and we can maybe deduce what that tells us about the rest.
Regarding Schopenhauer, the issue there is that the individual is limited - we only see a piece of reality, not the whole thing. For Schopenhauer's purposes, it matters not whether the universe is actually infinite as such or merely a googolplex of light years across. The point is that we'll never get to see all of it. We can never complete it; we can never win the game, and sit back satisfied at our Ultimate High Score. We just go on and on, fighting the slings and arrows of outrageous fortune, until we eventually and inevitably lose the fight and die. Our 'vanity' is our urge to see it all, to know and experience it all, to solve all problems, to survive all dangers, to live happily ever after. To imagine that we can. That the universe is effectively infinite from our finite perspective is more than sufficient to prevent that. That we'll never find out if it's actually infinite is just the sort of frustrating obstacle Schopenhauer was complaining about.
It might be worth noting that Schopenhauer in "Immortality: A Dialogue" talks some more about the relationship between the individual and the universe. Here he takes a panpsychist-sounding view that individuals are finite immanent pieces of the transcendental whole. As individuals they are finite: they will fail and die. But that 'Will to Live' they experience is shared by everything that exists, individuals feel it because they are parts of the universe, and when seen in that way, the Will to Live can succeed and we get to explore everything and live happily ever after. He's talking about the relationship between the individual and the universe. Whether or not the universe is actually technically infinite doesn't really make a difference. What matters is that the individual is only a small part of the whole.
