### 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.
}
	root->set_as_audio_listener_3d(true);
#endif 
	
	set_multiplayer(MultiplayerAPI::create_default_interface());
	root->set_as_audio_listener_2d(true);
	current_scene = nullptr;
	const int msaa_mode_2d = GLOBAL_DEF_BASIC(PropertyInfo(Variant::INT, "rendering/anti_aliasing/quality/msaa_2d", PROPERTY_HINT_ENUM, String::utf8("Disabled (Fastest),2× (Average),4× (Slow),8× (Slowest)")), 0);
	root->set_msaa_2d(Viewport::MSAA(msaa_mode_2d));
	const int msaa_mode_3d = GLOBAL_DEF_BASIC(PropertyInfo(Variant::INT, "rendering/anti_aliasing/quality/msaa_3d", PROPERTY_HINT_ENUM, String::utf8("Disabled (Fastest),2× (Average),4× (Slow),8× (Slowest)")), 0);
	root->set_msaa_3d(Viewport::MSAA(msaa_mode_3d));
	const bool transparent_background = GLOBAL_DEF("rendering/viewport/transparent_background", false);
	root->set_transparent_background(transparent_background);
	const int ssaa_mode = GLOBAL_DEF_BASIC(PropertyInfo(Variant::INT, "rendering/anti_aliasing/quality/screen_space_aa", PROPERTY_HINT_ENUM, "Disabled (Fastest),FXAA (Fast)"), 0);
	root->set_screen_space_aa(Viewport::ScreenSpaceAA(ssaa_mode));
	const bool use_taa = GLOBAL_DEF_BASIC("rendering/anti_aliasing/quality/use_taa", false);
	root->set_use_taa(use_taa);
	const bool use_debanding = GLOBAL_DEF("rendering/anti_aliasing/quality/use_debanding", false);
	root->set_use_debanding(use_debanding);
	const bool use_occlusion_culling = GLOBAL_DEF("rendering/occlusion_culling/use_occlusion_culling", false);
	root->set_use_occlusion_culling(use_occlusion_culling);
	float mesh_lod_threshold = GLOBAL_DEF(PropertyInfo(Variant::FLOAT, "rendering/mesh_lod/lod_change/threshold_pixels", PROPERTY_HINT_RANGE, "0,1024,0.1"), 1.0);
	root->set_mesh_lod_threshold(mesh_lod_threshold);
	bool snap_2d_transforms = GLOBAL_DEF("rendering/2d/snap/snap_2d_transforms_to_pixel", false);
	root->set_snap_2d_transforms_to_pixel(snap_2d_transforms);
	bool snap_2d_vertices = GLOBAL_DEF("rendering/2d/snap/snap_2d_vertices_to_pixel", false);
	root->set_snap_2d_vertices_to_pixel(snap_2d_vertices);
	// We setup VRS for the main viewport here, in the editor this will have little effect.
	const int vrs_mode = GLOBAL_DEF(PropertyInfo(Variant::INT, "rendering/vrs/mode", PROPERTY_HINT_ENUM, String::utf8("Disabled,Texture,XR")), 0);
	root->set_vrs_mode(Viewport::VRSMode(vrs_mode));