I think 5e can be deadly if you run the monsters right.
If it's a wild animal (i.e. owlbear), a hit by another PC will divert its attention. Otherwise, it may get in another hit on the downed PC.
If it's a battle with sapient monsters, they probably won't go after downed PCs because there's a battle going on and downing PCs who are still up and fighting is more important that killing already-downed PCs.
If it's a BBEG, s/he very well may kill a downed PC. Especially to further the "story". In my last session, a dragon, given a choice between downing 2 more PCs with a Wing Attack (which would result in 3 downed PCs), instead go for a killing blow with a Tail Attack. I did it to show just how cruel the dragon was. The dragon was running low on HP and was going to want to cut and run in the next round. So, leaving the party down but, frankly, unscathed probably would not have had a big impact. Now, there's going to be a session where the party deals with a dead-for-real PC.
If you have a problem with PCs not going down, adjust things. Use negative HP so that a downed PC doesn't get to 1 HP from just any healing. Scale up your encounters. Play your monsters a bit more ruthlessly. Add encounters or interrupt rests so that the party isn't rolling around with full resources in every encounter.