I have a couple bg I can find for this but If you’re ok w bl I have a few more
Im dating a psychopath(webtoon bg)
https://www.mangago.me/read-manga/black_chain/
Been a long time since I have read this one so not for sure if mc is suicidal https://www.mangago.me/read-manga/confined_together_with_the_horror_game_s_male_lead/
And im pretty sure the mc in this one goes in and out of suicidalness eventually just becoming numb https://www.mangago.me/read-manga/the_broken_ring_this_marriage_will_fail_anyway/
I spent an embarrassing amount of time looking through other bg I have read to find those but it was slim
If you’re ok with bl,
https://www.mangago.me/read-manga/death_delayed_for_a_short_time_because_of_the_will/
https://www.mangago.me/read-manga/the_hero_wants_to_die/
https://www.mangago.me/read-manga/stranger_2/ They both were suicidal at different points super sweet story though
https://www.mangago.me/read-manga/red_mansion/ Im not sure if suicidal is the right word but mc is crazy
https://www.mangago.me/read-manga/la_meche/
https://www.mangago.me/read-manga/payback_1/ This one, it seems really stupid for the first 30 or so chaps but if you get to the end it is so fucking beautiful the words said to mc when he is finally able to properly grieve
Ok this is all I can find for right now, I know there is more and sorry I couldn’t find more of the bg if that’s what you were looking for. I genuinely looked for over an hour because I also absolutely love this trope. Hopefully this helps scratch the itch a bit though!
I found a few more right after I stopped looking
https://www.mangago.me/read-manga/i_m_a_villainess_can_i_die/ Read this a long time ago completely forgot thr plot but I know she was very ready to die
https://www.mangago.me/read-manga/the_villainess_wants_to_die_gracefully/ This one is really weird
https://www.mangago.me/read-manga/a_gloomy_delusion/ This is a reincarnation bl
https://www.mangago.me/read-manga/again_1/ He killed himself in his first life
https://www.mangago.me/read-manga/father_i_don_t_want_to_get_married/ She struggles w feeling needed in both her first and second life
https://www.mangago.me/read-manga/sweet_shots/ Evb is worried abt this guy
ok im heading to bed now this took my energy out

reco manhwas where the mc is extremely suicidal and everyone else gets really really worried? ps. except kill the villainess.