You can.
Murder as a solution to problem stems from the origin of the hobby, but it's not set in stone. As a DM, you're empowered to strike down sociopathic characters. You address several different problems.
1. The emotional link between player and character.
Where you see thrill with the danger of dying, some players aren't keen on having their character die. They invest a lot of time and effort in creating a personality for their character and playing it right in their mind, and all this effort is reduced to ashes. They don't really seek the thrill of risking death, but the thrill of escaping death. When you go see a James Bond movie, and in the first 15 minutes of the film there is a firefight, you're 100% sure that James Bond won't be shot and become a cripple forever because the bullet struck too close to the spinal chord. You know he'll barely survive and that's enough. Some players want their characters to survive, not actually risk dying. In some groups, the DM doesn't kill a character without player's approval, prefering situation where they are captured, left for dead in a dire situation and so on. Maybe it's something that should be discussed on session 0 next time.
2. The prevalence of violence and murder as a problem-solving tool in RPGs
This also, you're empowered to change, and not necessarily by dropping the D&D framework. There are a few things you'd need to establish, before deciding if changing system is the way to go. First, I think it's a good idea to narrow down the problem with violence you're encountering.
a) Make violence tolerable
Our societies don't eschew violence, they eschew illegitimate violence. There are very few people who objects to unarmed or non-lethally armed police forces performing arrests, or to the existence of military forces to protect against an invader. If you're feeling ill at ease with the "kill them and take their stuff" approach, that is either glorified or expected in many published material, you might try to create a set-up where the characters are empowered to use violence within a certain context (members of the city watch...) and made accountable. And define what your group is OK with to create the "rules of the Watch". If they aren't into torturing suspects, forbid it. If they aren't into killing, force them to deal non-lethal damage and arrest bad guys instead of killing them, or have them act in self defence, mostly by prohibiting them to INITIATE violent action.
b) Reduce the drawback of not using violence
In many settings, it's complicated to take prisonners. OK, so the Medusa has surrendered and is totally OK to serve a period of community work like cobbling for six month as a punishment... Too bad that half of the village will be turned to stone while the Medusa cobbles the main street... But killing because it's convenient isn't acceptable ; so it's your duty as a DM to reduce the situations where the logistics of not-killing are so burdening that the more expedient solutions of killing people will be prefered by characters (and players, who don't want to hear that the evil wizard performed his world-ending ritual while they were ferrying back a few goblin guards from room A3 to the city for a trial). There are solutions to lessen this burden and incentivize sparing enemies, some of which can fit thematically a setting. It's easier in a mostly settled setting than between to far-away Points of Light, but it can be done. So characters will have the opportunity to use all the cool things in their fighting kits and face the risk of death (even if they know they'll win in the end) and you won't have necessarily to change system.
(This is assuming you're invested in the system and actually want to keep it. If you're just looking for another system, then it's easier to make the choice, and changing system will emphasize the change of playstyle to your players (especially if the above advice is applicable to your new world.).
