Can we interpret Rotations in 3D, about a fixed axis as simultaneous rotation about each component axes?

For a body starting from origin $(0,0,0)$ at time $t=0$ , and moving with a linear velocity $\vec{v}=v_x\hat{i}+v_y\hat{j}+v_z\hat{k}$ , after time $t$ , it reaches at $(v_x,v_y,v_z)$ . We know that it can reach same position even if the displacements were made along each axes ( one from $\{v_x\hat{i}, v_y\hat{j}, v_z\hat{k}\}$ ) at a time, in any order. But for rotations, orientation of the body rotating with angular velocity $\vec{w} =\omega_x \hat{i}+\omega_y \hat{i}+\omega_z \hat{i}$ cannot be necessarily reproduced if rotations are made about each axes ( $\{\omega_x, \omega_y, \omega_z\}$ ) in our choice of order (I understand it could happen if angular displacements are infinitesimally small). Now, there is another way to look at this. For linear motion, I guess everyone agrees that in each second the body moves simultaneously along x, y and z axes, with magnitudes $v_x, v_y, v_z$ units respectively. What about rotations? So, my question is the following: Can we explain rotation with angular velocity $\vec{\omega}$ in a similar fashion? - like simultaneous angular displacements ( $\omega_x,\omega_y,\omega_y$ ) are made about each axes? I had trouble understanding rotations in 3D for long time,even for a simple rigid sphere. But recently it helped a lot to know that, for a given fixed angular velocity $\vec{\omega}=\omega_x \hat{i}+\omega_y \hat{i}+\omega_z \hat{i}$ , it is taken as rotation about fixed axis in the direction of $\vec{\omega}$ ( i.e., unit vector $\vec{\omega}/|\vec{\omega}|$ ), with magnitude $|\vec{\omega}|$ rad/s. But is there still another way to look at this? - if so, I guess it is not simply $\omega_x,\omega_y$ and $\omega_z$ . Is there an existing transformation from these $\omega$ -components to such instantaneous angular displacements about each axes? - first of all, does it make sense to think about it like this? because maybe we cannot define angular displacement about one axis alone, when other angles are also changing? To further clarify, I am talking about, let's say a rigid sphere rotating without any external torque. I accept rotation about a fixed axis, so angular momentum is conserved. I have seen the questions question1 and question2 . In question1, confusion was why axis keep changing - it is easily understood clarifying it is not torque free rotation. In question2, combining two angular velocities as rotation about a new axis $\vec{\omega_1}+\vec{\omega_2}$ is clarified. My question is about a torque free rotation of a rigid sphere, which has a specified with angular velocity given by $\vec{\omega}=\omega_x \hat{i}+\omega_y \hat{i}+\omega_z \hat{i}$ .