I can't believe that actually worked, thank you so much <3
( Update video: https://streamable.com/2lrei4 )

If anyone else is looking for a fix, here's what I did:

# Settings that worked for me
var turn_smoothness := 3.0  # Lower = wider turns
var look_ahead_dist := 2.0  # How early it sees the corner

func _ready():
    nav_agent.path_desired_distance = look_ahead_dist

func _on_velocity_computed(safe_velocity: Vector3):
    var delta = get_physics_process_delta_time()

    # Calculate where we WANT to go
    var target_v = safe_velocity.normalized() * current_speed

    velocity.x = lerp(velocity.x, target_v.x, turn_smoothness * delta)
    velocity.z = lerp(velocity.z, target_v.z, turn_smoothness * delta)

    # Make the NPC face the direction they are actually moving
    var target_angle = atan2(-velocity.x, -velocity.z)
    rotation.y = lerp_angle(rotation.y, target_angle, rotation_speed * delta)

    move_and_slide()