And that's the "original" emotion, it's not filtered through another human being. When you watch a movie without audio and with subtitles (like in a bar or on a bus), the emotions of the speakers are already awfully clear from the visuals.
No, you can’t; how much emotion is shown via those things vs. tone, volume, and other auditory cues varies from speaker to speaker and speech to speech; sometimes, speakers demonstrate one emotion through gestures but indicate that it is insincere/being mocked/etc. via vocal cues, even.
Not to mention the degree to which simultaneously tracking face and subtitles makes you likely to miss parts of either or both.
For example, if we ask a question, it's not just that our voice goes up at the end. Our eyes move in a certain way too, slightly more opened and our eyebrows and sometimes cheeks raise.
And nobody's at a distance, the cameras are always on either a medium or close-up shot when filming politicians speaking.
If subtitles were equal value or even “good enough”, they’d be used exclusively. That they aren’t should tell you something, and you repeatedly protesting that you are unable to comprehend the value doesn’t mean it isn’t there.
Separately, relatively dry sarcasm can't be visually picked up directly, but a signer may be able to suggest some of that with body/hand language.
I don't know how you'd possibly attempt to objectively quantify that degree, but my guess is that you're understating it. The entire deaf community is probably not mistaken about which means of communication are the most effective for them.
Yes, but a deaf person has trouble hearing how the speaker is speaking. They're missing out on the emotion in the voice.
Consider all of the emotion that can be conveyed in an audiobook, or on a phone call, or through music. Humans convey a lot of emotion in sound that is not represented visually. Part of sign language is conveying the emotion usually present in speech.