> Death doesn't need to come at any particular time, but it does need to exist, looming just around the corner.

It always would: fatal accidents would still be a thing. So would:

https://en.wikipedia.org/wiki/List_of_Islamist_terrorist_att...

Then there'd always be the risk of a gigantic asteroid hitting the earth.

Stuff like that.

Which makes me wonder: if there was no more aging and no more illness and accidents and terror attacks / crimes were basically the only way to die, how would society deal with those?

I take we'd focus on preventing accidents / safety even more? For at the moment there's definitely some "we're not going to live forever anyway, so it's just bad luck if an accident happens".

And what about suicide? Taking your life when you're going to die anyway is one thing, taking it out when you're near immortal is something else altogether.