Transporters have to run on the quantum level resolution for organic beings to survive a transport. Quantum is the smallest you can go in this scale, so basically it is like reassembling that LEGO castle one piece at a time.
What I mean here is, in all probability the transporter does not
analyze the subject to quantum level, then destroy it, then build a new one out of those instructions and another batch of raw materials at the other end. It does not
use quantum resolution at all.
Instead, it turns existing structures into "phased matter" streams, a process that is more akin to taking a man-sized clump of the LEGO castle and placing it on a trolley than to breaking it down and carrying it by hand. The matter is not torn down to its constituent pieces, it's only transformed to "phased matter" which carries a gigantic amount of information with it, making it unnecessary for the transporter to handle that information.
If it were
possible to use the transporter to scan the subject down to the last detail and tear down and rebuild him, then the transporter could make infinite copies of the most intricate things. But this is not what we observe: at the endpoint of the journey, the phased matter stream is absolutely needed, or no reconstruction can take place. It is not just a case of sending across some abstract information for generic assembly of further Datas - it's a case of sending the original package, in a different, "phased" form.
"Second Chances" also seems to confirm that raw matter/energy can be used to recreate an object from the transporter data. It's not as if Thomas and William Riker were created from the same lump of matter that was the original person. The second containment beam must've contained an amount of raw material from the Potemkin.
But that was a highly exceptional situation. And if raw material from the Potemkin
were used, or used twice as much as normally, all sorts of klaxons would have gone off and immediately revealed to the world that two Rikers now existed.
Using raw material at the assembly spot is a very unlikely method for the Trek transporter to work, because typically the assembly spot is in the middle of nowhere. It is perfectly possible to beam Data into stark vacuum, for example. Or Lore, at any rate. The materials for that simply must be tagging along, not being replenished from any specific source.
So where did the material for Thomas Riker come from? It's not as if any of the laws of conservation of energy or matter would really hold in the Trek universe; the exotic energetic phenomena in the atmosphere might quite well have created duplicate atoms out of nothingness in a freak accident, and the universe shuddered a little and perhaps uncreated a few atoms elsewhere to compensate.
Then there's "Unnatural Selection" where a gentic experiment giving engineered "children" an active immune system leads to Pulaski and others contracting a virus that caused rapid aging by mutating DNA. Pulaski was "cured" by running her through the transporter and "filtering" her with a sample of her DNA pre-infection thus making her young again. (Well, "less old." )
However, in this case, there is no evidence that Pulaski's memory wouldn't have been reset in the process. It's not a rejuvenated Pulaski, it's the Pulaski from a few days back.
That's not something one would wish upon oneself: technologically induced dementia where you lose everything past your last backing-up date when you are rebooted, and have to be brought up to date by your (no doubt rapidly dispersing) friends.
The "Rascals" case is very different in that our heroes retained their memories while in the juvenilized bodies. That's eternal life and eternal youth for ya all right. But it wasn't done by the transporter. It was done by the freak spatiotemporal anomaly of the week. Without the anomaly, the transporter probably couldn't inflict that sort of specific "genetic damage" (actually, more like selective reassembly).
And the remedy Crusher and O'Brien devise need not be much different from the one done on Pulaski - except Picard seems to retain some sort of a memory of the events, as he's concerned about his re-lost hair and comments on how everything suddenly looks smaller. Doesn't really mean Picard would retain full memory, though - he might still have essentially lost the past few days.