"We must respond personally. The Universal Translator would be recognized." - Pavel Chekov
There is no doubt that this scene is mostly played for laughs, and seeing Starfleet officers frantically searching through books and speaking Klingon badly is an entertaining sight. But it's problematic if you think it through. At the same time, Chekov's statement highlights a significant issue for the Universal Translator.
First, some of what makes this scene difficult to accept. To start with - and this is really just nitpicking - why does Chekov get the disembodied voice over about the UT? Wouldn't that be Uhura's line?
Would a Starfleet vessel really have a library of books about the Klingon language? Would Uhura really speak Klingon like a gringo? Hoshi Sato was probably spinning in her grave.The Klingon translators at Kirk and McCoy's trial speak better English than she speaks Klingon. (Supposedly Nichelle Nichols objected to the scene because she believed Uhura would have been able to speak better Klingon.)
So there are some oddities here we just have to be willing to accept for the sake of the gag, but we have to ask, how could this little ruse have been at all convincing? If the Universal Translator would be recognized, what about Uhura's poorly spoken Klingon? Bernd Schneider's article "Alien Monocultures in Star Trek" offers an intriguing possibility. It suggests that not all Klingons necessarily share the same native language and continues: "Maybe this was even the reason why Uhura's awkward attempt to speak Klingon in 'Star Trek: The Undiscovered Country' did not raise any suspicion."
In any event, what does Chekov's statement mean? In what sense would the Universal Translator be recognizable? Evidently he wasn't talking about not replying with proper pronunciation and grammar, since they didn't do that anyways. And he evidently wasn't worried about the delay in translation, since their human efforts to translate certainly took longer than the UT would have taken. So it must primarily have been a concern over the quality of the synthesized voice.
We might assume that the quality of the UT's voice would improve over time, as the technology improves in general and as it became more familiar with a given language. The question becomes, would a 23rd century UT really have such trouble with Klingon when there had already been extensive contact between these two races? By this time there were individual Klingons that were fluent in English. They had an ambassador that could address the Federation Council and the Federation President. They sent a whole group of delegates to the Khitomer peace conference.
But the problem of the Universal Translator being recognizable over an audio channel only multiplies with face to face communication. The obstacles we have identified so far would only compound: Beyond the basic challenge of providing an accurate and nuanced translation and dealing with a potential artificial sounding voice, there would necessarily be a delay and lip-syncing issues.
'The Universal Translator would be recognized' indeed.
So how is that Starfleet officers - even of Archer's time - could seamlessly intermingle with alien populations and not be picked out as impostors?
Even if we accept the seemingly magical ability of the UT to learn alien languages instantly and basically perfectly, such a device would always be recognizable while in use. Storytelling convention allows us to gloss over this for the most part. We don't have to be subjected to the delays, the strange sounding voices, the lip-syncing issues every time. But then by slight-of-hand we slip from storytelling convention to simply ignoring these issues altogether.