A lot of BotW's appeal is how hands-off it is, especially in this era of modern gaming where absolutely everything has to be dotted, marked, made to glow, talked about, signposted, and otherwise given to the player for the sake of ease of difficulty and guidance. BotW instead tosses you into Hyrule and tells you to go fix it, or play in it, with the least amount of hand-holding or back-patting from Papa Nintendo as possible, which aides the feeling that you yourself are on a true-blue adventure. Just about every decision you make in the game is of your own volition, not at the command of a designer, and it gives the moments players have had with the game- the battles, figuring out puzzles and recipes, finding hidden mechanics, experimenting with the physics, engaging with NPCs, or just sitting on top of a mountain and immersing one's self in the audio and visuals- a certain feeling of authenticity that is refreshing.
BotW has a very distinct atmosphere and life to it in that regard that would be utterly infringed upon by unnecessary prompts like achievement pop-ups for arbitrary collections, kills, dungeon and shrine accomplishments, or- at worst- simple things like escaping the Shrine of Resurrection. Imagine that- that moment where Link gets out of the Shrine and runs up the hill, the leitmotif's piano begins stirring, the camera pulls back to reveal the whole of Hyrule, and then *Boop* "ACHIEVEMENT UNLOCKED: FREEDOM! PRESS START TO LEARN MORE!" It would utterly ruin the moment. Hell, people don't even like the fact that the HUD can't be completely free of icons related to health and stamina. I myself can't go back to anything less than the Pro HUD after finally trying it and am almost in the camp of wishing the hearts and stamina meter would go away as well.
Achievements wouldn't "hurt anything" in that they would actually make the game mechanically worse. But the nature of the way they're commonly implemented actively goes against a lot of BotW's intent and the subsequent joy it has wrung from its fans, and I feel some of my most memorable moments and accomplishments with it would be somewhat marred by their inclusion.