Now.. I imagine that all this analysis in itself seems a bit disrespectful of human emotions, treating them as just moving parts in a mental machine. And you're right. Emotions are not "just" moving parts in a mental machine, at least not in the sense that many people would interpret that sentence. Most people reading that would interpret it to mean that we should care about emotions roughly the same amount as we would care about some random moving part inside a machine. Now some rationalist could try to say that moving parts inside machines really are important, and so these other humans shouldn't get all fussy about someone saying that emotions are "just" moving parts, because that's actually a great honor. But the whole issue is about whether the speaker really holds emotions in the same high esteem as the person hearing the statement. That is, if you think that I "don't care" about emotions, then you might be weary of me, since you might suspect that I'd have no problem doing something that injured your emotions, which you certainly would care about, even if I didn't.
I care about emotions. I care about other people's emotions. I also think that emotions are "just" part of the way the human brain is designed, and are computationally comparable to something we might find in a relatively simple AI algorithm. However, I am a human, and so I am subject to emotions. I also live in a world with other humans who are also subject to emotions.
I feel like my views on this raises the question: "Do you feel emotions?" Yes. I feel emotions very strongly, and I value and embrace a great many emotional experiences. I get emotional listening to music, and I love listening to music for that reason, and I find that some of my best thoughts are coupled with strong emotions. I even nurse a bit of sentimentality at times, and other feeling of grandeur looking at beautiful scenes.
Also: "Do you like it when people analyze your emotions and try to tell you to just get over it?" No. I value and legitimize my emotions, and it pains me deeply when people belittle them. I don't get angry, and I try to maintain a constructive conversation with even the most ill-intentioned people, but a part of me wants to say "fuck you."