I'll note that Emi suffers from PTSD
, and worse, is in a country that so strongly stigmatizes mental health treatment that, most likely, the only KS character actually getting treatment is Hanako.
Basically this. Different people react differently to tragedy and feel differently about it, (and oftentimes don't have control over it) and I think it's unfair to say that she's exaggerating because you don't know how she feels. She isn't your sister*. She's Emi. Which you've acknowledged, but I don't think you've considered why that might be enough.
Addressing the aforementioned inconsistencies first. When she kicks Hisao out of her house, she does it because in that moment it feels to her, depending on your choices, either that she has been betrayed or that she is being pressured. There is more than one way to hide feelings. Being stoic is one. Getting angry is another. For Emi it's the latter, because while she actually does puts up a cheerful facade as a first defense to try and hide her feelings, (which leads to perceived inconsistencies) she doesn't know how to handle somebody seeing through that and doesn't have much of a backup plan. It doesn't seem to happen often, and it suddenly forces her to relive what happened and makes her feel pressured to let somebody in, which can be very scary and leads to an outburst. Her method of hiding doesn't account for what happens when somebody realizes she has a problem and how to hide her feelings then; she doesn't know what to do then and doesn't think that far ahead. It's not even in her nature to think far ahead- notice that she puts down pirate as a career choice specifically because of that. Her strategy for hiding her emotions is instead making it seem like there isn't a problem at all so that there's no need to even worry about the later part. This is why the switch seems so sudden and out of nowhere- Hisao wasn't supposed to catch on to her issues in the first place. He was supposed to keep pretending everything was all happy and good, and he didn't. Her exaction then was a panicked knee-jerk reaction, in this case, anger, most likely because she's a fireball in all aspects of life, which tends to be a double-edged sword at times. It's not really inconsistent with the rest of her personality, it just looks that way because there's a lot going on behind the scenes.
This is amplified by the fact that all of this is taking place near the anniversary of the event and she seems to get especially upset during that time, even faking being busy with finals to try and avoid getting too emotional about it around Hisao and trying to avoid giving him a clue that something is off in the process. An old interview with her writer, Hivemind, even states that she tends to feel worse about it than usual around that time. That isn't melodrama, it's emotion. Saying "x person has seen worse but acts better about it" doesn't change how she feels or make her feel any better. In fact, that's like saying "these things happen" which explicitly doesn't help her. Some people who are starving get along just fine in Africa without expressing much emotion- does that suddenly solve all of your problems and change how you feel about them? It doesn't for me. Comparing emotions for different levels of stress and different levels of tragedy and struggle and acting like they should equate on some mathematical scale just doesn't work. Emotions aren't rational like that.
As for her being inconsiderate, I'd say this is also untrue. She does use Hisao as relief to an extent, but she doesn't lead him on because that implies that she never wanted to get closer to him to begin with and only plans to toss him aside, which is false. Like I said, she does want to move on, but comes to a mental block when the emotional aspect gets to a certain point. She doesn't just play around with Hisao's emotions, she uses sex as a coping mechanism while she struggles to find a way to let him in. She doesn't totally fail to consider his feelings either. It doesn't look very much like she does because of her sexual coping mechanism, but a good part of the reason she is able to finally come through at the end and trust Hisao with her story is because she does realize how much he cares about her and she has thought about that. She wouldn't struggle so much and do something so significant for somebody she didn't think gave a damn. Reread the scene where she begins to tell the story, and notice how slowly she has to build up to telling Hisao even when they're already at the site. It's painful. She had wanted to that for a long time, but she needed a push to get there that was a happy medium between firm and understanding, which in the good ending, Hisao manages to find.
She does have problems, just like everybody else. Serious, damaging, hurtful ones. She puts up a shield of cheerful denial and acts rudely when it is pierced. Her inability to move past paper-thin sexual endeavors as a distraction is not a positive. She can be stubborn to a fault, and often is. She's notably vulgar. Her short-sightedness hurts her at times. She doesn't play up her feelings for attention though. She isn't disjointed, and she isn't weak. Even with her occasional temper flares, the vast majority of the time she is kind and caring. She works to improve herself each day, and has done so significantly by the end of the VN's school year, more than a lot of people do in a lifetime. She gets there with Hisao's help, but Hisao does not force her to get better as a person- she still has to do that in the end because nobody in the world can do it for her, and I think she does a pretty good job of it, all things considered. She owns her mistakes and tries her best to move on and learn from them, and that's all you can do.
*My greatest sympathies go out to your sister, by the way. I by no means intended to minimize her issues or struggles with this comment, and I think it's good that she can deal with her emotions on a more controlled level.