Because that would ruin the fun.
Bad response, but kinda the point. The show has to keep up drama, which can be hard in a post-scarcity future; "why don't you just spend money/energy?" sounds like a great solution.
There is one bit I'd wished they did use the transporter for...
According to the Technical Manual, the transporter has a "bio-filter" that can filter out known and unknown harmful bio-signatures from the transporter signal.
It's literally humanoid antivirus! It scans the incoming signal for telltale bio-signals indicating disease life forms and deletes them.
(I always thought it'd be interesting to have somebody run to the bathroom after a transport because the biofilter was set too high and stripped out her gut flora...)
But that points out exactly why the bio-filter isn't a cure-all. It's a first line defense against plagues being brought onboard, but the concept of a "disease organism" is a bit hard to define. e coli on the skin is fine; e coli in the intestines are doing it's job; e coli in the blood is a very dangerous situation. And I doubt the bio-filter can differentiate... here's why:
In my answer to the question Does transporting (beaming out) leave your troops vulnerable to enemy attacks?, the Technical Manual tells us that the "matter stream" in the transporter is an ANALOG signal. Dialog from Scotty and O'Brien heavily imply that the transporter is almost completely analog; you can boost it, filter it, twist it into knots. It's why there's a continuity of awareness while in transport... you never stop being a thing, you just go through a conversion process.
So, the bio-filter can scan for well known "danger signals" and filter them out; it can identify a few "nasty" critters and block them out, but it probably doesn't have the fidelity to carefully snip individual E. coli from specific regions of the body. Transport only takes 5 seconds, a process which is mostly continuous, with only a pattern buffer to delay transport.
Scotty could use a pattern buffer to stay ALIVE, but he couldn't make himself younger... or fix his arm.
The amount of data in a human shaped blob of particles is YUUUGE. You're not likely going to be able to keep it in computer memory (another reason transporters are analog; only a buffer is needed, as usually the transport is nonstop). A single human would require more memory than... well, a lot of memory.
So what the transporter does is grab a "template" scan of the transport subject, and uses that scan to "error correct" the transport buffer while it's being filtered and stored. Analog signals tend to degrade over time; the template exists to "fine tune" the buffer to keep a signal "coherent." It's why we're so sad that Scotty's red-shirted friend doesn't make it; too much signal degradation. (More on this thought below!*) So over the years, Scotty's signal was amplified and then re-shuffled with the "template" to keep things in line... and what came out was almost, but not quite, what had originally gone in.
Remember, the scanner takes a look at NOW, stores a compressed version of that, and uses the template to error correct. But it's not a "complete trace."
It's why you can't beam over a pile of bricks when you lose somebody in the beam and just slap their stored template on it. You'd get something which is almost, but not quite, entirely un-alive. It's error correction, not the data itself.
In Rascals, the transporter was able to use an older copy of the "error correction trace" to re-construct the kids as adults. I hate this episode, as a fan of "transporter tech," but that's basically how it works. They beam over from the shuttle, there's a malfunction, the error correction system tries to make everything better but ends up making the main cast kids. They do kid things and then get blended up into particle soup again to reconstruct the adults from old templates.
Note that the adults do NOT remember what the kids did... this episode LITERALLY murders four kids. They were created in the transporter, and they died in the transporter, because the transporter needed their matter to apply the "error correction" template on.
It's possible that the error correction template is some sort of machine learning; maybe the main cast have gone through the Enterprise teleporter that it's really, really good at fixing problems when the plot light isn't lit.
And then using DNA to fix somebody... aaargh So my theory, based on what I've read about transporter tech, is that the intrepid crew were able to beam the patient out and then MODIFY the error correction data using her DNA to remove the disease; when they beamed her back in the error corrector kicked in and "retemplated" the Doc into working order. It's a bit of a stretch, and a very bad way to resolve the problem (because of this EXACT question being triggered by it), but there you go. Writers gonna write.
Merging two people into one, and then splitting them up... GARAGH! The second half makes sense; throw in the original particles as Tuvix, blend him up into particle paste, and then use the error correction filters to spit out the separate crew members. The initial merging, well, that's totally because, if you look right here you can see OH MY GOD LOOK OVER THERE TOM PARIS IS TURNING INTO AN INTERDIMENSIONAL TRANSWARP LIZARD!
Needless to say, to keep my sanity, I try to ignore Voyager whenever I can.
Long rambly answer short: the transporter would never be able to save all the quantum data from all the crew members on board for transporter reconstruction. You'd need some sort of "duplicate" in stasis to use as a template, and that's just creepy horror story time. Instead, the transporter was designed to convey an object via an analog "data stream" so that the object is never actually "disintegrated." It's, instead, converted to energy and converted back. TOS transporters used big heavy duty amplifiers to punch signals through noise. Later transporters added layers of "error correction" on top of the signal stream, which allows you basic filtering of bad bio-data and pattern correction before reintegration. The error correction "fudges" the signal after amplification, filtering out glitches and smoothing over bugs, attempting to give you a clean signal. In the process, a small bit of data, the error correction "transporter trace", is stored in the computer. In general, a trace is only useful for THIS SPECIFIC TRANSPORT, as using an old trace would inevitably "reconstruct" an older version of the passenger, wiping out any changes which had happened in the intervening time. Wouldn't that have been nice for the new Science Officer in Star Trek: The Slow Motion Picture? But by Movie 6 and TNG, transporters had this trace. "The more complete the trace, the more "error correction" could be overlaid on the incoming signal to fix issues during transport, and some clever people in the TNG era figured out how to use that to "inline edit" the matter stream. Of course, nothing in life is free, and this sort of meddling is EXCRUCIATINGLY DANGEROUS. The only time the bio-filter dares step in is during an obvious infection being transported, of a signal it doesn't recognize, in fears of scrambling the subject. And laying a trace on top of somebody, at best, may restore the structure of the patient, and at worse (in the case of Rascals and Tuvix) murder the passengers beaming out to "restore" a backup of the original passengers.
(* The death of Scotty's Friend. I always thought this was unnecessarily melodramatic; a cost saving measure to not have to pay for another speaking role for a whole episode. Unless Scotty built his "regeneration device" to prioritize his own signal over his friend's, why would two signals, held in the same buffer, suffer different levels of degredation? I always thought the story would have been told better if Scotty's friend had stuck around as a reminder/balance for Scotty... Scotty is stuck in the future, everything is strange, and all the future people think he's a fossil, but at least Red Shirt thinks he's cool. But that's me...)
