This is a complex question, and answering it will involve adressing several aspects of it.
What did Schopenhauer believe, based on his quote?
This is your first sub question, and Schopenhauer is not here to answer. But we on PhilSE have tried to answer this question previously in this link: What does Schopenhauer mean by 'A man can do what he wants, but not want what he wants'?
Note, none of the posted answers agree with your interpretation of Schopenhauer. The top answer holds that Schopenhauer considered determinism and free will to both be imperfect categories, and our world and actions are a confusing intermingling of both.
All of the other answers note the different tiers of desiring, or of willing. They note it is true that we cannot choose whether we have the desire for food or not, but can choose whether we have the desire to act on the desire for food, and can choose not to act. The tiers of desiring or willing do not appear in your interpretation.
But if one cannot "will what one wills", where is any choice?
Schopenhauer is not the last word on this question, and the meat of what you want to ask is in the bold above. Whether that is a misunderstanding of Schopenhauer or not, is irrelevant to your more substantive question.
I note that the "tiers of desire, or willing" interpretation does not address your bold, as the determined/random dichotomy can then just be imposed on the higher tier of our will to action.
I have also encountered Virtue Ethics thinking that our higher tier tends to be determined in practice, but AFTER making what we realize is a mistake, we then CHOOSE to restructure our character, so as to avoid that mistake in the future, pushing freedom up to a higher tier yet. This too, does not evade a determined/random dichotomy.
Ultimately, libertarian free will requires rejecting a determined/random dichotomy, for at least one tier of willing, as the first answer on the prior question about Schopenhauer implies.
The determined/random dichotomy is an empirical assertion. It is a claim that this logic limits what our world can exhibit. Note that logicians have realized there are infinite different logics. Whether a particular logic matches our world or not, is an empirical question. The need for a third option, that of agent causation, to apply in our world, is just a description of the logic form that is needed for libertarian free will to hold. With infinite logics in logic space, we will be able to find one that supports three options, where the third matches reasonably well our understanding of agent causation. I don't think we have yet found that logic yet, but Agent Causation theorists have been searching for it.
Is the position of many who disbelieve in free will that our experience of making decisions is illusory?
Yes. For free will incompatibilism, anyone who is a free will denier will say we do not make decisions.
Free compatibilists will not deny free will and will say we make decisions. But they will redefine both free will and decisions such that they do not match the libertarian assumptions.
Is the ability to make decisions an incontrovertible fact?
No. There are no "incontrovertible facts". We could be mistaken about our willing, about our experiencing, about our selfhood, and about the state sequencing of time.
BUT -- these are all VERY basic observations. Empiricism relies upon observations, which we draw inferences from. These observations are the base data of empiricism. Most of these -- have instances where we are mistaken about our base observations. That is part of being human, we are fallible, even in these very basic perceptions. BUT any argument that we should throw out whole categories of our basic observations, rather than a few erroneous instances, carries a massive burden of justification.
This applies to Einstein's claim that the "now" of time is not real, to Dennett's claim we have no experiences, and Dennett and Gautama's claim that there is no real self, and to your claim that we have no will.
In each case, that of Dennett, Einstein, Dennett again, and you here -- the denier of experiences has a THEORY, which the experiences appear to contradict. Rejecting the data of experience, in favor of a desired theory -- is directly contrary to the scientific method.
The answer, for an empiricist, is: look harder, and find a theory that encompasses all the data.
For Einstein, it is to try to find a theory of time that includes both block time, growing time, and presentism. For Dennett, it is to find an ontology that accommodates experiences, and selfhood, as well as our physical world. For you, it is to develop a theory of causation that allows for agent causation.
OR, if you could put a strong enough evidenced case together that there can only be one valid theory of causation, and determined/random is it, AND the experiences of willing are so unreliable that one should dismiss them, then you could reasonably argue for your POV. I have seen no effective presentation of such a case, and the history of our revisions to causation theory, and its current weak characterization today, suggest such a case is impossible.
Your disputant's response: "the assertion that we cannot make decisions is absurd", is not particularly helpful. Rather than "absurd", it is instead in contradiction of basic observations, and is a prioritization of theory over data, without the accompanying very strong case that such a prioritizing requires.
