Most of the time I can't stand when people use this greeting to me, but not for exactly the same reasons as the OP. Like it's just extremely hard for me to answer with a standard "I'm fine, and you?" sort of response if I'm not actually doing well. I just really have a hard time trying to pretend that I'm fine when I'm not. It actually hurts emotionally to try to do this.
For a long time I tried to answer honestly and say that I wasn't doing well when I wasn't, but then people would actually want to know why I wasn't doing well and I'm not sure if they really wanted to know or if they were just being polite, but then I'd find myself getting drawn into longer conversations about all the crap going on in my life with people I didn't want to be having these conversations with such as cashiers, bus drivers, or even casual acquaintances.
Then for awhile I tried answering with "I'm alive" because it was the only positive thing I could say without lying. This generally backfired because that phrase seems to send a huge red flag to people that I'm really not okay, and then they'd still ask what was wrong.
These days if it's not someone that I want to be asking me that question, I've kind of settled into ignoring them altogether. The other person might say "Hi, how are you?" or sometimes skip the "Hi" part for some reason. "Then I'll just respond with "Hi". Oddly enough, some people will repeat "How are you?" if they don't get a response, and I'll just say "Hi" again, which does feel awkward because they don't seem to understand that I heard them the first time but I'm just choosing not to engage in that particular social convention, but it's still preferable to me than either lying or getting dragged into conversations that I don't want to have.
The only time I don't mind being asked is if it's someone that I feel cares about me and actually wants to know how I'm doing. And that's really the only time that I'll ask the question, if I actually care and want to know how they're doing. I also make an exception for medical and mental health professionals asking because it's their job to care how I'm doing and if I don't talk to them about how I am, then its a wasted visit.
The last good friend I had was for awhile in the habit of asking me how I was a few times a day. I know a lot of people would find that annoying, but I actually really liked it a lot. It made me feel like someone actually gave a damn about me and really cared about me for once in my life. Most days I don't get asked how I'm feeling even once by someone that I feel actually cares about me. So it really sucked big time when she stopped doing it. I know it was more because her life was in a really bad place at the time and she was too wrapped up in her own problems than because she stopped caring about me, but it still hurt a lot. It especially hurt since I was in a really bad place too at the time and still took the time to check in and see how she was, even though listening to her was often detrimental to my own mental health. This eventually led to the friendship falling apart, but that's another story.
So yeah, that question just really sucks as a social nicety and I don't want to participate with people that I don't care about. I also find the phrases "What's up?" and "Sup?" to be really irritating for some reason and wish people wouldn't greet me with them at all. Even though I know exactly what the other person means, because the phrases rub me the wrong way, I can't seem to stop myself from answering them extremely literally. For example, I might respond with "The ceiling" or "No thanks, I'm not hungry" This usually either gets a laugh or confuses people, but I've actually had people try to explain the slang to me. Yeah, I don't really care. The majority of the time I only want to be greeted with "Hello" or "Hi".