First the minor problem, which concerns head-canon (i.e. personal canon) interpretation of the TOS phaser dematerialize setting. So, it was said in dialog that dematerialize causes things to vanish without a trace. Cool, that's what I always thought: it just transforms matter into pure energy (cf
The Making of Star Trek page 193).
A few frames later and the inderite sculpture test target is turned into vapor by super-heating. Wait, what? Vanish without a trace ≠ vaporize. Vapor is matter in the gas phase; to vaporize = to cause to evaporate (or sublimate). And what's that smoldering pile of debris left behind, if the dematerialize setting causes things to vanish without a trace?
But it gets worse. When the pirates are melting the door with their phasers, they are breathing in fumes of super-heated metal which they can smell and that makes them cough, as liquified metal flows towards them that their phasers continue to only vaporize in an enclosed space. By conservation of matter (since here phasers only vaporize), there are literally tons of super-hot metal vapor particles in that room at that point. The pirates themselves should be melting and evaporating/sublimating. Anything with an autoignition temperature below a few thousand degrees (pick your temperature scale; that includes clothing woven from organic material such as cotton, it probably includes hair, but also it includes those regular metals such as lead and iron that react exothermically with oxygen, and the list goes on) would be a combination of on fire, melting, and boiling, at least until the breathable air runs out at which point the pirates start suffocating (as if they could still be alive). If they had put on environmental suits, it would have been much more OK, at least except for the part that the whole room should have melted/vaporized/caught on fire/etc.
Eventually also that vapor is going to cool, leaving literally tons of traces....
But it would all have been OK if phasers just turn things they dematerialize into things that aren't material, i.e. energy composed of photons but maybe also neutrinos, antineutrinos, and/or other non-material or nearly non-material particles (either of the exotic
Trek-y variety or otherwise) that don't interact with matter under normal circumstances, basically whatever combination necessary not to give the people standing in the same room a flash of fatal radiation (e.g. gamma rays). And that would make phasers much scarier, since aside from a barely detectable radiation pulse that dissipates into space, nothing at all would be left, no trace whatsoever, except perhaps a pattern of dematerialization left where the edges of the phaser effect were, basically as Mudd said in the first place ("no trace").
There's just no way to square the term "vaporize" any other way.
Still, this is just a minor problem. As I said, I can imagine that something else happened on the tech side, as indicated, forget they coughed, forget they smelled fumes, forget they were endangered by an expanding pool of molten metal, and the problem in the story just goes away.
The more serious problem is the fate of the pirates at the end. T'Chok just lets them go without a memory wipe. Um, what? She is the leader of a group of ideologically motivated activists (no doubt considered radicals) willing to have their own minds erased to protect their cause. Why would she just let them go, knowing what they know? Perhaps this is setting up a future story, but honestly, it makes no sense, as is.
I did however thoroughly enjoy the idea that there is a group of scientists turned activist erasing their WMDs from Federation science.
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