Both options have pros and cons; in practice I'd advocate for using a mix of techniques, as appropriate for the situation.
When the animation drives the object's world position, the animator has significantly increased control over the artistic look and feel of the animation and how realistic (or stylistic) they'd like it to be. But the game has significantly less control over the object, leading to more work and/or potentially very ugly looking stopgap techniques to (for example) prevent objects from being forced inside other objects due to the animation (if collision is off). Or if collision is on, preventing the animation-driven displacement from being countered by collision, destroying all the animator's hard work as the object "slides" against some surface while animating. IME it's better to let the animation drive the object for complex interactions like clambering over obstacles.
When the game drives the object's world position, the game logic has much more precise control over the object; since it's usually game logic that translates player input into displacement of player-controlled objects, this can result in much tighter-feeling control. But it can also result in less than ideal visuals because the animators typically cannot know for sure when the game will blend between two animations, and some blending techniques can result in obviously-video-gamey transitions in scenarios where, for example, you stop a character short in their walk cycle. IME it's better to let the game drive the position for locomotion related actions in particular, and also as a default choice.
There's a variety of ways to combine animation techniques in a mix-and-match approach that let you hide or minimize some of the cons of either approach (motion matching, for example, is a Relatively Hot New Thing recently).
What I recommend you specifically do is look at what is most efficient for you: is it easier for you to work in Blender and have all your displacements authored there? Or is it easier for you to do displacements programmatically (for the majority of cases). Either way you'll likely need to put work in to smooth over artifacts, so you'll be paying that cost. So look at which option will let you personally pay the cheapest cost for your project's needs.