The thing is, transporting isn't described like this in the Trek universe at all. Notably absent are any expressions suggesting "taking apart", except when McCoy huffs and puffs. In contrast, we see people put through the first stage of transporting, turned from visible and tangible to invisible and intangible - and then continue existing as an integrated and interactive whole until moved elsewhere and/or turned back to normal.
Well, a medical man should know. And being converted into energy, beamed somewhere, and then converted back to their "original form," does strongly suggest your constituent matter parts are no longer being held together in the normal fashion at some point, anyway. There is also a notable change from
Wrath of Khan onward. Prior, one was immobile and had no real sense of the passage of time, though in
Mirror, Mirror Kirk claimed to get dizzy in transport, but mostly they did little and moved little if at all and experienced nothing mid transport. And whatever sensation they got in the transporter buffer may have been imagined. After
TMP, they seemed more able to move, even conduct an uninterrupted conversation, and by
TNG they could interact and grab things in some in-between zone. Inconsistencies are most likely attributed to never really knowing how the tech works and individual writers taking whatever liberties they wished to make a story work.
I wonder what Scotty was thinking for those 75 years he was in the pattern buffer.
This would make one question why the use of a receiving transporter is so uncommon in practice. When there is no receiving platform, the choices would be a) beam down anyway and b) don't beam down. And the latter ought to always win if receiving platforms really are significant for safety.
Not really. Taking a shuttle down or crossing the road is not without risks. Transporting to an area without a receiving transporter just means if something goes wrong, the odds are greater you will die. Normally, nothing goes wrong, and such a transport may be (in fact some say is) safer than most alternative methods of getting there. A receiving transporter just would add an extra layer of backup and protection, as well as take less energy (perhaps) and safeguard against unwanted visitors (microbes, hostile Klingons, weapons, explosives, for example) by providing a safe and open conduit but only into a protected area.
We have ample evidence that once turned into a transportable form, the victim can be handled much like a UPS shipment - rerouted, temporarily stored, sent into an address that no longer exists and bounced back, kept going even though the sending city was flooded out of existence, perhaps even digitized for one stretch of the voyage. Transporting where two machines keep constant vigil over the package is at most possible, but clearly not necessary nor commonplace. Speculating on an operating principle that in fact relies on two machines watching over it all is fruitless; speculating on one that benefits from a second machine should be done with moderation.
I don't know how many examples, if any, you can dig up of the sending transporter mechanism being destroyed before transport was complete, but the transport being completed anyway without the aid of a second receiving transporter to finish the task, but I don't think there should be any. That doesn't mean there aren't, but none come immediately to mind.
Transporter tech is daft and problematic, of course, as are many other Trek technologies (replicators, holograms, artificial gravity, etc), and arguing about their inconsistencies as if there is one true and correct way they "actually work" is almost pointless, beyond entertainment, and entertaining some ideas about how it might work or what they problems might be.
Memory Alpha suggests (though who always know from what stories they derive these ideas, or if it comes from elsewhere) that your so-called package can hang around in a pattern buffer for 420 seconds or 7 minutes before it will start to degrade. Scotty, of course, found a way to stretch that to 75 years, but not reliably, and Janeway, IIRC, was hiding telepaths in the pattern buffer for relatively short inspection periods, but even that began to cause health problems, so in that way it's not too much like a UPS shipment, unless you mean getting a package that was damaged in transit happens more than you'd like.
No doubt a transporter signal is a suitably weird phenomenon that allows for odd geometries, such as seeming "spreading" or "dodging" or whatnot. Perhaps an advanced transporter might even push its beam through the Earth. But Gary Seven is a good example of transporting not being about machines, but about a natural phenomenon: no matter who builds the machine, another machine built for the same purpose (by aliens from across millennia!) will be able to handle the package. This undermines the idea of a "signal" consisting of data organized by the cultural quirks of the user, as such would be impossible to handle by another culture. Unless we insert UT technology into the transporter, that is...
Despite some minor cosmetic differences that may be introduced at the beginning or end, if it is a transporter beam, or a compatible transporter beam, it probably replies on the same scientific principles and therefore any machine capable of handling those can read those signals since they are not cultural or arbitrary. There may only be one way to represent an electron in such a way, or a neutron or proton, or their relative positions and velocities. If some cultural oddity is possible or is introduced, then an alien system won't recognize it, or won't be able to handle it. And Gray 7 had a machine – a transporter mechanism, both on the homeworld thousands of light years away and in his office walk-in vault.
Two factors cast serious doubt on that. One, the sender is supposed to stop sending to avoid the calamity. Two, the sender is supposed to boost his signal in order to get the package delivered intact.
Whatever the receiver is supposed to do, the technobabble on that is actually almost nonexistentially thin. Which is natural in a sense - Kirk would look silly giving himself voice commands, even if he's doing crucial things with his fingers. But the fact that Kirk gives crucial-sounding voice commands to the other end, while that end has nothing to say to Kirk, suggests Kirk isn't an empowered party in this affair.
If another system (starbase transporters) are handling it as the active system, then the malfunction on the Enterprise should be irrelevant. The starbase would beam a person to the ship just like any transporter beams a person someplace and rematerializes them at the target destination sans assistance.
In a two-transporter hand off, the first system could just finish the job if the second one failed, or vice versa. When two transporters are involved, the whole idea is one can take up the slack if the other fails. Therefore I do not think they were using a two-transporter hand off in the way they probably should have been.
What I think happened was the Enterprise was using its own transporters without assistance from the starbase (which is maybe not atypical, but if I had backups, I'd use them). A malfunction occurred. Engineering tried to alert the transporter room before they engaged, but were too late. Rand had started beaming two people aboard. She tried to stop it but couldn't. Rand requested the starbase to
"Override us. Pull them back" mid-transport, showing definitively it was the Enterprise that had the active system. "Us." Kirk then took over after Rand requested the starbase transporters try to grab the signal and put them back together. Kirk suggested to starbase transporter control boosting the matter gain, as captains often do by assuming one more expert in their field needs his help or suggestions, but sometimes lesser mortals panic, and Kirk does not panic by nature, so even if they are already doing it, there is little harm in suggesting or encouraging that course of action. He fights for more signal, or laments they need more, but they don't have it.
Had Kirk been more expert, doubtless he would have cross-circuited to B, but he wasn't, and he didn't.
But the thing is, this isn't about tech. Phasing is a natural phenomenon, and phased packages exist regardless of the exact machinery involved.
Yes, the matter stream exists, sort of like how a photon of light exist as it travels ever onward under its own steam, but how long the matter stream can do that before its signal degrades (7 minutes) is short, and it can't just rematerialize itself "back" into its original form without tech assistance that I know of. Although, Chekov felt those Klingons they beamed into the pattern buffer were "non existent" and should remain that way in
Day of the Dove.
But you list so many good ones! Kirk is mildly annoyed that the transporters aren't working. He isn't in any hurry, though - he takes his sweet time taking the deliberate long route not just with the little space pod, but with the Turbolift as well. And he smiles all the way to the Bridge.
Kirk got exactly what he wanted, it seems. And the first half of the movie is all about that: the world may go up in flames, but for Kirk it's just an excuse for a good time.
Kirk wanted a change, sure, and regretted giving up command, but to even suggest he didn't care about the world as long as he got what he wanted is just as unlikely that he was happy his girlfriend died on the transporter pad since it saved him the trouble of breaking up with her. Kirk is not like that. And he was in a hurry. No reason not to relish the moment, but that moment was fleeting.