Nevertheless, tremolos are done with the wrist and your fingers are simply aiming for the right keys.
That's a very extreme way of putting it. This approach only works if the fingers are already highly developed. My tremolos were poorly controlled and sluggish for years, with this approach. If you don't engage the fingers, there's a major limit to speed. It's just that those who have highly developed finger technique don't focus on this aspect. If you don't have that, this approach can be truly disastrous for some. It held me back for many years.
Also, the wrist has no way of getting involved in anything much to do with tremolos. It just needs to be loose. You cannot rotate from the wrist- it comes from the elbow and acts upon the whole forearm. The wrist is just a joint in the middle. This may sound like a mere technicality, but I'm not just being pedantic. The more you understand any rotation to stem from further back, the easier it tends to be. Thinking at the wrist (straight away) can sometimes cause forearm stiffness, unless the action is already very well evolved.