I think the more important question here is why do you want to be more okay with PC death? When Gary Gygax ran his games back in the day, he was the original Killer GM. Everything in those old dungeons has a save-or-die effect, and that was rolled into how the players approached the game (ie, with backup characters and not a huge amount of attachment to individual PCs).
But how you handle character death in your campaign is really a social contract between you and your players. If your players are not excited about the idea of dying all the time, why make them? I know you're playing 5e, but the Fate Core rulebook has some really good system-agnostic advice on character death
Most of the time, sudden character death is a pretty boring outcome when compared to putting the character through hell. On top of that, all the story threads that character was connected to just kind of stall with no resolution, and you have to expend a bunch of effort and time figuring out how to get a new character into play mid-stride.
That doesn't mean there's no room for character death in the game, however. We just recommend that you save that possibility for conflicts that are extremely pivotal, dramatic, and meaningful for the character -- in other words, conflicts in which the character would knowingly and willingly risk dying in order to win. Players and GMs, if you've got the feeling that you're in that kind of conflict, talk it out when you're setting the scene and see how people feel.
Basically, character death should always be a major event, that leaves the player feeling like it was worth it.
An example from one of my games:
For a year or two, I ran what appeared to be an extremely deadly campaign. The characters were always facing things a couple CRs too high, and barely escaped most encounters with their lives. But over the course of that campaign, only one character actually died. His name was Rimble, and he was a ranger.
Earlier in the session, Rimble had been humiliated by a rival ranger who'd gotten the jump on him. From that point on, he was actively looking for a way to redeem himself. The climax of the session was a battle in a secret thieves-guild hideout with a giant were-rat trying to smash his way into the room from below. The rat had made a big hole in the middle of the floor, and the rest of the party were wisely moving their fights away from the hole.
But not Rimble. No, not Rimble.
"NOW'S MY CHANCE," he screamed, and charged towards the hole in the floor, to leap at the rat.
Now, I knew he was going to die if he faced the rat head on. Its CR was so high that it was more of a level hazard than an actual opponent. But I wasn't sure he knew. So I asked him,
"Are you sure you want to do this?"
"Oh HELL yes," said Rimble's player. But I still wasn't satisfied. I told him that in order to get to the were-rat, he first had to jump across the fire pit in the center of the room. He rolled the check, and got a natural one, pitching headfirst into the fire. I figured that would be the end of it.
But next round, Rimble picked himself up, brushed the embers off of is face, and screamed
"NOW'S MY CHANCE!"
Then he leapt through the hole in the floor and onto the were-rat's face, wielding his mundane shortsword. He died that round. Everyone was in fits of laughter, overcome by Rimble's valor and stupidity. A few rounds later they found silver weapons, and were able to chase the rat off with a deluge of colloidal silver, which beautifully outlined Rimble's shattered corpse where it still clung to the rat's face. There were a lot of memorable moments in that campaign, but that was probably the best.
And the important bit was, Rimble's player was just as happy as everyone else. At the end of the day, no matter what system you're running, you're collaboratively telling a story with your players. You're not playing against them. Even a killer GM is a killer GM because that's what the players signed up for.
So basically, the best way to conquer your fear of character death is to make sure that your PCs are ready to die.
But if you still want to get deadly with it...
That said, if you're not having a good time because you don't feel like your games are deadly enough, but you don't want to piss off your players, the best advice I can give is to get a better feel for how close you can come to killing your party, without actually killing them. While not everyone enjoys losing their characters, everyone will fondly remember a scenario where their character almost died.
This is what the unconscious/unstable rules are designed for. Unless your monsters are dealing massive damage, once a character drops to negative HP and becomes unconscious, they've still got a pretty good chance of surviving, due to death saving throws (without help from anyone else in the party, the quickest they can possibly die is in three rounds). These saving throws are generally pretty tense moments for the players, and usually make for a memorable session.
If your players aren't interested in losing their characters all the time, you're never going to be able to escape pulling your punches altogether. For example, you probably shouldn't include save-or-die effects in your game, or abilities that you know have a high probability of killing characters without giving them a chance to react. But it sounds like you can probably go a bit harder than you are currently. Provided you are putting them up against level-appropriate challenges, the system will actually be surprisingly forgiving.