Of course they can. Normal doesn't really mean nothing bad ever happened to you, just that the things that happened to you don't rule over your life. I never had my parents die in front of me, but a good amount of people have seen people die in front of them and that doesn't have to ruin the rest of their lives for them. Same logic applies to any sort of trauma more or less.
Was the accident very traumatic for Ami? Most certainly. Was the fact that she has received absolutely no professional help, nor a stable home or dependable adult also a problem? Yes, and I would go so far as saying it is even more problematic than her parents dying. Her parents dying was an event that was deeply traumatic, as was when Miku's parents died - but the aftermath was a continued event that went on for years, hasn't stopped and put such a strain on the mind of a kid that it warped her psyque much more than just them dying.
In other words, by not dealing with it in a proper manner, it's like her parents are still dying. Both her and Akira are, in some level, still on that day, they never left. The fact that Akira calls her daughter and she calls him dad is not just important for their interpersonal relationship, but also, in my view, because it starts to manifest the reality that Nozomu is dead. They form a proper family, they grieve and cry together, they bond over their shared trauma and they take control over their life again despite what happened. It would have been much better to do it then, but it's never too late.
Regardless, this is all a minor point - Ami is the way she is because things were done in the way they were. We'll all just continue hoping she can get as healthy as she can now that Akira's taking responsabilty.