Some of these arguments are completely absurd, and some of them are self-contradictory, but that's never stopped people from believing them regardless. Writing centers are unethical because the tutors who work there write the students' papers for them. Writing centers are unethical because they undermine instructors' goals. Teachers give writing assignments to evaluate individual student learning and progress. Assistance from a writing center tutor problematizes that kind of assessment.
Writing centers are unethical because they short-circuit a student's learning process.
Even if tutors don't write the students' papers for them, the help tutors provide gives students quick, easy answers to problems that the students should be solving themselves. This gives an unfair advantage to the students who use the center. Writing centers are unethical because Hillary Rodham Clinton used to work in one, and if she was associated with them, then they must be horrible places. Don't be surprised if someone, somewhere, thinks the writing center is unethical because it kills so many trees for its brochures and handouts. I'll trust each of you to handle such situations as they arise.