Your frustration in your first sentence I would equate to a general human frustration that we all feel when we see our own sins reflected in others -- you don't back up your claim so you get mad at me for not backing up my claim. I could in turn make a really good case that the burden of proof is not on the person who says "he doesn't make that argument" to prove that he doesn't, but on the person who says "he makes that argument" to prove that he does. And then we can go back and forth in negativity, I suppose.

In an attempt to avert that spiral of world-suck, let me attempt to instead meet your request head-on. Penrose is not saying what you say he's saying, and would not be able to fill up two books with your argument.

Let's talk hacker for a second. What you're saying is that Penrose is basically saying "well I don't understand this bug in my computer program, and I don't understand the kernel in my computer, so therefore a kernel bug is causing this bug in my computer program." You are going to be predisposed to see any reduction of a bug towards the kernel as being evidence that yes, your characterization of the argument is correct and the argument is indeed laughable. This is understandable if someone just told you that hypothesis off-the-cuff, which I surmise is how you've probably encountered Penrose. (That is, I take the fact that you're laughing as evidence that you've not read both The Emperor's New Mind and Shadows of the Mind.) The alternative however, is that someone has (a) a high-level statement of "this shouldn't be happening", and (b) knows that the only way that such things can happen, barring things like the MMU misbehaving, is a kernel bug, and (c) has been attempting to trace down the code to the fundamental syscalls, and found that the thing-that-shouldn't-be-happening is indeed bracketed around one syscall in particular which does not seem to be doing what the docs say it should do. And if a hacker has attempted all of this, I mean, they might still be wrong: but their argument is no longer something of the form "I don't understand the kernel and I don't understand this bug, therefore the kernel caused this bug." It is something more sophisticated than that, and you're doing a disservice by laughing at them.

One and a half of Penrose's books are ultimately about why the computationalist/algorithmist thesis of brain function seems deeply incomplete -- that a hallmark of some certain examples of reasoning appears to be deeply nonalgorithmic. (For example The Emperor's New Mind supplements his usual Godel argument with arguments that if brain plasticity comes from local neural plasticity then it is unlikely to have a great generality, but by a clever analogy if it does have a great generality then not only does that mesh with an idea of global plasticity, but it is very unlikely to be algorithmic. The one and a half books are basically full of little digs that, seen together, do seriously undermine faith in the thesis.) This is essentially the "bug" above. His favorite example is that we are capable of proving to our own mathematical understanding that our mathematical-understanding-algorithm contains true sentences which we will never be able to prove; if this algorithm exists then it is able to prove something that it will never be able to prove, which seems like a straightforward contradiction. But there are one and a half pop-sci books that he's written building up some other problems with this thesis in a way that even if you don't start off with knowledge of the relevant physics/comp-sci/whatever, you can hopefully understand it enough to appreciate the problem.

He follows these up with a principled dive into what's actually underlying these "bugs". That is, he discards ideas about souls and takes it seriously that our reasoning is realized in our brain, and takes seriously that this is a biochemical system, and evaluates whether the noncomputable phenomena that he is "debugging" can come from chaos or randomness or the like. He is able to rule most of these out with auxiliary considerations, and so tries to get one level closer to the hardware: the biochemsitry is well-modeled, we know, by physical chemistry; there are likely no new fundamental interactions or whatever to be found in biochemistry.

Looking into the physical chemistry level he finds that it can be somewhat cleanly divided into two parts: quantum and classical. Technically, it's all quantum: but you can handle a lot more with various classical rules of thumb, and this has basically all been done before him so he can just steal those results:the classical part of physical chemistry and probably even the vast majority of the quantum part of physical chemistry are algorithmic, it is known that quantum computers don't do other things than what computers do; they just might do many things more quickly. There exists only one part of the quantum part of physical chemistry which is not.

Unfortunately, he doesn't have access to the source code, nor docs for what nature does--so he actually constructs physical-chemistry models that could hypothetically do the sort of buggy things that he's seeing at the top-level that he has carefully ruled out at the other levels of explanation.

And this is all to say: as you can hopefully now see, these books contain a lot more than simply "we don't understand X, we don't understand Y, therefore Y causes X." Namely there is an actual understanding for example that X is realized in a system which has to be modeled with Y, and which has some features that can only come from certain corresponding features in Y, which we actually do understand very well. There has been a solid amount of good effort to push down that causal chain until one is left at these syscalls and then saying "something is fishy about those." And his argument might indeed be wrong, there are many little parts where maybe he's wrong about how chaos works and how it could cause a similar bug, just as our programmer above might be wrong about whether two processes are properly synchronized via locks or whatever -- but it's no longer this laughable thing of "ha, he doesn't understand X and Y so he thinks Y causes X!" that you're trying to say.