The claim that critical skills are not essential for clear, fair-minded analytical writing as well as for good analytical reading is clearly false, as critical thinking skills are crucial for success in both.
Analytical writing is, by definition, writing that requires analysis and logical reasoning. In order to do this effectively, the writer must be critical of what she reads. That is, she should read actively, asking reasonable questions and noting poor reasoning and inadequate or missing evidence.
Analytical writing requires the same application of skills, only from the perspective of the person making the argument, not considering it (so...using the same skills but in reverse). The analytical writer must consider his proposition (thesis) and assess what arguments are most persuasive in support of that proposition, then find reliable, relevant, and recent sources to back his assertions.