I mean, as you say, all of this can mostly be accomplished by simply calibrating your harm effects in a manner that seems reasonable to you and your fellow players. Conceptualizing it as adding a '0 harm' result seems solid -- after all, Lash Out Physically just says 'you cause them harm', and presumably the amount of harm is fictionally determined. There doesn't really appear to be any need to change the moves; just don't say something does 1 harm if it doesn't do 1 harm.
1 harm examples from the book include ramming someone with a car or pushing someone down a flight of stairs (though to be fair, they also include kicking someone in the groin.) I don't feel like it's really that far off from the world you are imagining.
Personally, I've never had this problem in the game, but in general my games don't tend to have a lot of low-level/incidental violence -- when violence does happen it is intense, and so people being mortally wounded easily has rarely seemed like a drawback. This is particularly true because the sort of default response to going to 4 harm is to enter your Darkest Self, which is something most players are excited about, so you really have to be taking a lot of harm (and doing so fairly regularly) before that sort of fictional switch starts to feel repetitive or forced. In my experience, the 'Darkest Self' reaction is especially appropriate for the sort of slow-accumulation-of-1-harm that you find frustrating -- where being kicked in the groin and thrown down some stairs and generally battered by the world may not lead naturally into 'and then you die!' it leads very naturally into 'and then you can't take it anymore and you become a monstrous Werewolf/evil Witch/hungry Ghoul' or whatever.
This contrasts with injuries that seem to fictionally justify the possibility of death, or at least serious bodily consequence -- when that kind of injury triggers 'death', the 'Darkest Self' out becomes less of a given, and other choices (including actually dying) naturally become more likely.
That said, I can see how the Harm track actually secretly being a 'trigger your Darkest Self eventually' countdown clock as well as a 'maybe you could die now if you want' measurement could be frustrating, especially in a game where violence is occurring more regularly.