The Chinese room argument is just missing kinda one basic premise and people don't realize it and argue around the premise without actually addressing it.
“Can we agree that of the many different things folks mean when they say your consciousness is ‘causally generated by your brain,’ that at least we can say your brain in its present state is only capable of generating exactly one conscious process at a time, namely you?”
So it's a sort of causal sufficiency, Searle believes that there is something about this squishy machine that the physics is doing that makes it conscious, and that's being caused very directly by the squishy machine. It's not some magic.
On the other hand, if you believed in souls, you might think that a given brain contained both the soul of a human person, and the soul of some demon possessing them—two conscious processes, one of which was stuck “along for the ride.” [Searle is also interested in the cases of split-brain patients where you have a similar phenomenon because essentially two separate brains inhabit a single body. He has I think mentioned at a talk that one of the interesting things about consciousness to him is that it all gets Unified, so it's interesting that if the two parts of the brain can talk to each other they merge their consciousnesses into one more powerful consciousness, rather like (my analogy, not his) how if you have two water droplets on a plastic plate and you push them with a toothpick together, at some point they merge into one bigger drop.]
Now the Chinese room thought experiment is not about the computational model of consciousness—not directly! It was always phrased as a rebuttal of the Turing test in particular, and the computational model only indirectly after the Turing test falls. Note that the Turing test has no direct beef with the casual sufficiency axiom, which is why it went unstated originally. According to the Turing test, written text goes in, written text comes back out, a dialogue appears to happen to the outsider, this is sufficient to conclude conscious understanding of the language used, which confirms consciousness.
Searle’s objection is, “if it's really all about inputs and outputs and not how I get it done, then you've left out what for me feels like the most important part about understanding a language: understanding it is part of how I get it done!” Right? You understand English because you can phrase your ideas into it, you can mold it to suit you, and it can (when heard) so mold you—it’s not just because the words can come out of your mouth, triggered by other words that came in through your ears. The Turing test has always ever stated “don't worry what happens in the middle” and Searle is saying “but for me understanding is part of the details that are happening in the middle!”
So where does the computational theory come in? It comes in because Searle wants to make this argument rigorous! He says, “if your computational theory is true, then there is in fact another way that I can speak a language—words in, words out—where I don't understand a word of Chinese. I am Turing complete, I could memorize a program for speaking Chinese and happen to execute it flawlessly and at no point would my ideas get into Chinese, at no point would the things that I heard mold me. So the Turing test is crap!”
The prominence of the words “I, me” is what invokes the idea of causal sufficiency here, “I have the right inputs and outputs, but I don't understand.” The Turing test is not able to distinguish between multiple consciousnesses instantiated by the same hardware, if such is even possible. The inputs and outputs go into the same box, as far as Turing is concerned as long as only one conversation happens, there's only one person in there.
This does do a great job of defanging the Turing test, because all of the ways that you might weaken the concept to address this major limitation do make it sound completely tautological. “Yeah well something consciousnessy is happening in that box but I don't know what.” / “Okay then why are we even talking about it.” / “Because computers can speak!” / “Right, so we care that computers can speak because they can speak?” / “No, like, we gotta give them rights now, or some shit.” / “John Searle already has human rights, if he's memorized a program that lets him speak Chinese without understanding it, you're saying we need to give that program human rights?” / “Yeah!” / “So uh is it murder if Searle decides that running the program is no longer fun? Is he a slave to this program forever?” / “Uhhhh...”
It doesn't defang the computational model, not directly. But the computational model does imply that VMs exist. We use them every day! And that's all that the Chinese room is, it's running a VM inside of another computer, one consciousness carrying a separate consciousness inside of it, a willful sort of demonic possession. The only thing the Chinese room has to say about this, is that we don't use our language very well if it is true. Philosophers who believe in that will need to generate an alternative language that is able to distinguish between “I am doing it” and “I am sustaining a daemon who is doing it,” because for them that's a real difference, you might have a hundred consciousnesses in your head that you don't have direct access to. That is a necessary part of believing that consciousness is software, you don't know if you're in a VM inside your brain, you don't know if something you're doing is actually secretly a Brainfuck program instantiating another VM inside your consciousness, software embeds within software, that's a core feature of software.
But of course Searle thinks that that's kind of ridiculous because he thinks that it's obvious that consciousness is something that the squishy wetware of the brain does, and this forces him to believe in that causal sufficiency—“my brain is only sustaining one consciousness, namely me,”—which the computationalists cannot ever agree with because that's not how software works. Anyone who believes in that causal sufficiency, even if they don't have the same basis that Searle does for believing in it, also thinks that the computationalists are ridiculous.
But the point is, that's happening at a level way above the Chinese room argument, Chinese room baseball came and went, now this is a whole separate game being played at the same ballpark afterwards.