The Transporter scans you to the deepest quantum level
Isn't that supposed to be impossible in quantum mechanics? Flippancy aside, you're assuming the transporter works like a computer scanner of sorts, "digitizing" a pattern at one end and reintegrating it at the other end. Suppose it's a wormhole where the subjects are "pushed" through subspace whole? One could make that argument, as Kirk saw the transport technician attacked in "That Which Survives" while he was in the process of transport. (The same with Kirk and Saavik talking
during transport in
The Wrath of Khan, although I understand if one might argue that is not "canon" being outside the series.) Still, one might argue that Kirk (in "That Which Survives") was in the process of being scanned, in which case the process is not "instantaneous" and one would have a mental "hiccup" every time one was transported. Yet Kirk remembered what he saw on the other side.
It's a rabbit hole many have gone down in these forums—how does the transporter work?—and anywhere the nuances of
Trek are discussed. Speculate if it makes you happy, but it's a lot of pointless effort, especially with TOS which was still feeling out the sort of fine-grain continuity the argument seeks. I'm sure one could find similar discontinuities in all the
Trek spin-offs, despite having been through the sifter of precedent stories, or a detailed writer's "bible."
And then there's the zeitgeist thing. Many older sci-fi stories have a "World War II" mindset, with individually piloted fighter craft "dogfighting" in the vastness of space, despite having engines that could blip one out of an enemy's gunsights in the "wink of an eye," etc. That sort of dogfighting is rare today. And weapons launched by today's fighters are of an entirely different class. There are countless sci-fi stories that imagine weapons of war far beyond even the "capital ships slugging it out" that is popular in the
Trek universe. The point is, the raw computing power to do what is suggested in the original post—scan and buffer such a massive set of information, and with programming or AI to analyze it all and warn an operator—are completely outside the "mindset" of the technology at the time the episode was written.
(Historians call it "presentism," or judging the past by the values of today, despite the fact that
Trek was imagining a future embodying many sci-fi tropes. Now understand the task of prequel designers trying to make a new show look like it fits into the past of TOS. Touchscreens and the like in those older ships?)
For that matter, what about the opening scene in "Dagger of the Mind" where transport could not take effect until the Tantalus Colony lowered its shield? Think about it—they could readily communicate with Tantalus. Old style radio, or subspace, despite being close enough for old fashioned radio to have no appreciable latency? How does the transporter "communicate"? Is there a "firewall" in the Tantalus screen that allows one sort of communication through, but not another? (Consider that firewalling did not exist in IT until the 1980s.) Again, how does the transporter work? If it reaches through "subspace" or "hyperspace" in some fashion for its zero-latency function, how does one shield against that? That's like entering a completely doorless room by using a time machine to invade the space "fourth dimensionally." (Enter the space before it is closed off, then come forward.)
We must accept many things simply because that is how the writer wrote it.