People calling the effort you put in to support your partners, friends and family "emotional labor" are either blatantly misinformed or people who want a pass on not giving a shit about their "close ones".
Emotional labor, as a term, was created to explain the difficulties and effort someone has to engage in to regulate their emotions when they're constantly dealing with the suffering of other people during work. It's valid, just as long as you use it in its appropriate context. This dumbass appropriation of the term by a certain branch of liberals is like if someone used the physical concept of entropy to justify why they're never getting out of depression.
If someone only wants emotionless relationships with people they only interact with for their own benefit, and never giving a care in turn, that's legitimate, as long as they don't lie about their intentions. But that might also explain why this Hannah at the OP cannot find a good partner.
We all have mental disorders. Fuckin everybody does. Especially those considered normal. If you have no problems fitting into this corpo hellscape, you're nuts.
The irony about depression and entropy is that is actually a pretty good analogy. Depresson, just like entropy, will only cool down more and more over time. You quite directly have to put in effort to solve depresson just like 'something' has to be countering entropy for difference to remain. Entropy untreated leaves you with nothing to work with much like depresson.
It works well as a metaphor, which is why people might be fooled to think there might be a direct parallelism without understanding the insurmountable differences between both (depression may be very hard to get out of if you're in a downwards spiral, but definitely possible, while entropy is literally an inevitability of the Universe, as far as our understanding of physics goes) which is why I compared it to the popular appropriation of emotional labor, which notices the poetic similarity but is unable to understand the actual differences between both.