This is what I mean:
https://www.nature.com/articles/419883a traumatic experiences (as an example) are burnt into our memories. Other less severe negative experiences can also be burnt into our memories and we cannot simply erase these experiences (aka forget about them as if they never happened).
In the end it's learn to cope with the experience (with therapy, meds and other tools). That can work but there is no guarantee that it works. Especially when the person isn't actively willing to cope and therefore has given up and sees the only solution to it in suicide.
So your assumption on how trauma works is off. So with trauma the reaction gets trained. You don't need to access the memory to have triggers and reactions. Often you'll find those with PTSD and CPTSD unable to access the memory due to it being repressed. They still experience triggers, emotional flashbacks, nightmares, and the like.
Yes, traumatic incidents form strong memories. Touching the memory usually results in triggering a reaction. Understanding of PTSD has improved quite a bit since that paper came out.
The trigger and behavior however remains. Treatment involves reprocessing the memory and retraining the behavior. In the end, it's like it's just a bad memory. You don't see the drop in activity in medial prefrontal cortex and the like on fmri.
Coping tools help deal with symptoms in the interim and dealing with stress from therapy.
Feelings of not caring about getting better, beliefs of hopelessness, and suicidal thoughts are symptoms of depression. You're justifying beliefs of suicidal ideation due to hopelessness. They're separate. Even without the hopelessness, it wouldn't necessarily eliminate the suicidal ideation. There are hopeless ppl who don't feel suicidal. These should be understood as individual symptoms of depression.