Now, you’re getting the inflection of the interpreter and not the speaker (derivative). Certainly you’re not hearing inflection on the original orator, either, so maybe it’s a mixed bag.
Subtitles in general, and live captioning especially, avoid editorializing or taking creative license- they're literal translations of words without other cues added in.
One nice exception to this has been the community subtitling work involved in various Taskmaster spinoff series, where explanations for cultural jokes or idioms are added into the subtitles for foreign audiences, especially when the joke is a pun or relies on a mispronounced word to make sense.
Live subtitling (on the BBC at least) is mainly done using re-speaking and voice recognition, rather than typing.
On the other hand, signing involves actual translation, not just transcription, which is much more likely to drop meaning or introduce confusion. Translation is already hard enough, and live translation is a whole other level of difficulty.