Even counting only killing violence can come up in stories that aren't generally violent. For example, the Wingspan board game adaptation is great but could be seen as fairly violent considering the fish and mice shown as food, although even as a vegan I find that a quite different feel than the "shoot and kill stuff" games (although there is one more graphic card that is a bit disturbing). Of course that one doesn't have a story, so not really what you are looking for, but I find it relaxing to play music plus the bird sounds and it is a fairly quick game once you get used to it (AI is a bit slow, though).

I would also say something like Trails in the Sky, while violent in a number of different ways, has a quite different feel from FPS style violence and has an extremely extensive story (just the Sky series is likely 200-300 hours, so not the best recommendation when you have kids :/).

I avoid any kind of game with blood (unfortunately no blood options are not that common these days), although I find non-red blood much less disturbing. In addition to the story there is often a strategic element connected to the violence that and I appreciate games that try to do more complex stuff without violence. Card City Nights is one where there are card game battles instead of fighting (there are a few bits of the story that make it less enjoyable than it could have been but mostly it is wacky in a good way). If there is violence I much prefer it to be more cartoony and ideally not involve killing. The old SNES Donkey Kong Country games are great at that, not very violent to begin with IIRC and the violence is very cartoony.

Story could be considered in maybe four aspects: plot, world building, symbolism, and interpersonal interactions. IMO, games almost never have a decent plot. I find good world building and symbolism to be the most common.

Anyway, enough rambling and to repeat the above suggestions and more (with a note that my memory isn't great so I could easily be forgetting something):

Good less violent games with some kind of story:

* My Brother Rabbit - hidden object puzzle with minigames

* A Short Hike - platformer with flying mechanic and parkour racing

* Rime - finding stuff and light puzzles, heavy symbolism

* Little Bug - two-stick platformer again more symbolism heavy

* Card City Nights - wacky story with card game battles

* NyxQuest - another good platformer with unusual mechanics, minimal story

* SNES Donkey Kong Country series - excellent platformers with mostly environmental obstacles, minimal story

* Beatbuddy - music synchronized platformer, minimal story

* Calico - cat cafe and a magical island, lots of bugs but cute, minimal story

* The Witness - puzzle game, symbolism but not much story

* Botanicula - music heavy adventure game with cartoon violence

* Creaks - puzzle + adventure game with some cartoon volence

Good less violent games with no or almost no story:

* 140 - music synchronized platformer

* Timberborn - sim game with beavers

* Ostriv - traditional Ukranian village sim

* Wingspan - boardgame adaptation

* Cyber Hook - 3d grappling platformer

* Kerbal Space Program - can be quite violent if you aren't careful or don't retry when things go wrong :/

Not low volence but good story or setting:

* 8doors - gameplay similar to Hollow Knight with an afterlife setting inspired by Korean folktales

* Trails in the Sky - likely most extensive story in a game series with turn-based battles and some volent themes in the story

There is a DRM-free game store focused on story heavy games called Fireflower Games. I think it is run by one person in Sweden and has a small selection of mostly adventure games (and not necessarily low violence but generally less violent). I haven't purchased anything there (not a big fan of adventure games) but it might be worth a look.

https://fireflowergames.com

Everything I mentioned is available on GOG, although Calico isn't kept up to date there unfortunately.