I think the two main reasons are:
It destroys a person's future. Do you notice the way suicides are reported? A lot of the time, the person (apparently) had a glittering future ahead of them. Whether that's realistic is questionable. Maybe they were that gifted and talented. Presumably, people firmly believe that whatever was hampering them so badly to want to end it would eventually pass and they'd become an incredible success. Or, at least, happy with their lives.
I suppose, if they are pro-life, they likely experience life in a positive way and they presumably believe we can too- just as soon as we get past this mental block or illness. That's the other thing- 'normies' frequently blame ideation on mental illness. So- they see it as a cruel affliction rather than a reasoned perspective.
- Deaths in a family or of loved ones causes grief for those left behind. I think a suicide is a particularly painful experience for those left behind. It forces them to realise just how unhappy that person was when they were alive. So- along with the usual feelings of missing that person's presence, they may also feel guilt that maybe there was more they could have done to help them. I suspect some suicides lead to further suicides.
So- I wouldn't agree that it's just like any other death so, it shouldn't matter. We're emotional creatures. Even natural deaths are experienced differently. People will react differently to losing their 89 year old grandparents to losing a 10 year old from a sudden illness. Death isn't expected amongst younger people. Same goes for violent deaths- accidents or murders. There's a lot more shock and resentment there.
Suicide is complicated because in theory- it was that person's choice. That may be incredibly painful for a loved one to accept though. It's a rejection I suppose. They may have loved them and supported them but, it wasn't enough to make them want to stay. Plus, we do actually suicide knowing how much it's going to hurt them. It may be by default type thing. An unfotunate side effect to our main goal but they likely realise we considered them and did it anyway. I suspect that's too painful for some people to accept. So, they'd rather think their loved one wasn't thinking clearly. They'd probably prefer to see them as vulnerable and susceptible to somewhere like this, (or- how they see this place at least- as a death cult) rather than accept that their loved one made a reasoned decision where they did in fact consider them and what affect their suicide would have and chose to do it regardless.