### Godot version

4.1 dev3

### System information

Windows 11, Forward+, Intel 13700k, 4070ti

### Issue description

In 4.0.x and in 4.1 dev2, if you overwrite the rotation of a bone in `_process` it has preferred over the currently running animation. In 4.1 dev3, the running animation takes precedence and overwrites any changes in `_process`. This happens both with a simple animation and with an animation tree, and happens in-editor and in-game.

### Steps to reproduce

1. Open the attached project.
2. Click on `AnimationPlayer` and pick an animation, e.g. `Armature|Standing`. Set it to looping if it's not already. (@RandomShaper's note: and play it).
3. Click on the `PointOfInterest` and move it along the X axis.
4. If you're on:
    - 4.0.x or 4.1-dev2: Notice the head rotating
    - 4.1-dev3: Notice that nothing happens

### Minimal reproduction project

[BoneRotationIssue-4.0.3.zip](https://github.com/godotengine/godot/files/11582187/BoneRotationIssue-4.0.3.zip)
[BoneRotationIssue-4.1-dev.zip](https://github.com/godotengine/godot/files/11582188/BoneRotationIssue-4.1-dev.zip)


Should include all licensing stuff for the model, but let me just add [the link](https://opengameart.org/content/human-model-ready-for-animation) for good measure.
r_error.error = Callable::CallError::CALL_OK;
	ERR_FAIL_COND(p_argcount < 3);
	ERR_FAIL_COND(!p_args[0]->is_num());
	ERR_FAIL_COND(p_args[1]->get_type() != Variant::STRING_NAME && p_args[1]->get_type() != Variant::STRING);
	ERR_FAIL_COND(p_args[2]->get_type() != Variant::STRING_NAME && p_args[2]->get_type() != Variant::STRING);
	int flags = *p_args[0];
	StringName group = *p_args[1];
	StringName method = *p_args[2];
	call_group_flagsp(flags, group, method, p_args + 3, p_argcount - 3);
}
void SceneTree::_call_group(const Variant **p_args, int p_argcount, Callable::CallError &r_error) {
	r_error.error = Callable::CallError::CALL_OK;
	ERR_FAIL_COND(p_argcount < 2);
	ERR_FAIL_COND(p_args[0]->get_type() != Variant::STRING_NAME && p_args[0]->get_type() != Variant::STRING);
	ERR_FAIL_COND(p_args[1]->get_type() != Variant::STRING_NAME && p_args[1]->get_type() != Variant::STRING);
	StringName group = *p_args[0];
	StringName method = *p_args[1];
	call_group_flagsp(GROUP_CALL_DEFAULT, group, method, p_args + 2, p_argcount - 2);
}
int64_t SceneTree::get_frame() const {