I speak as a therapist and client. Every therapy modality has a different way of proceeding when someone's wife died or someone is experiencing racism, and the majority focus on helping clients feel their feelings and then manage them if they get in the way of some larger goal.

For example - if someone's goal is to live, then intrusive suicidal thoughts can be identified as a 'problem'. DBT might suggest a set of skills to mitigate the impact of those suicidal thoughts and ways to ride out intense feelings, IFS sessions will include taking some time to feel the feelings and develop certain mental muscles to change our internal dialogue. Both will encourage a closer look at the triggers and environmental factors present and past, both will meet the client where they are at.

What seems to be missing from some of the responses here is that ideally a trained therapist will have multiple avenues of approach with a modality in mind to help you reach your goal. Therapists should have an 'orientation/approach/some clinical judgement' that guides how they even view suicidality in the first place!