### 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.
i--;
					pg_count--;
				}
			}
			if (pg_count != process_groups.size()) {
				process_groups.resize(pg_count);
			}
		}
		{
			
			process_groups.sort_custom<ProcessGroupSort>();
		}
		process_groups_dirty = false;
	}
	
	
	
	uint32_t group_count = process_groups.size();
	if (group_count == 0) {
		return;
	}
	process_last_pass++; 
	uint32_t from = 0;
	uint32_t process_count = 0;
	nodes_removed_on_group_call_lock++;
	int current_order = process_groups[0]->owner ? process_groups[0]->owner->data.process_thread_group_order : 0;
	bool current_threaded = process_groups[0]->owner ? process_groups[0]->owner->data.process_thread_group == Node::PROCESS_THREAD_GROUP_SUB_THREAD : false;
	for (uint32_t i = 0; i <= group_count; i++) {
		int order = i < group_count && process_groups[i]->owner ? process_groups[i]->owner->data.process_thread_group_order : 0;