On the other hand, does not banning these debates contribute to having less of them?

The debates are not technically, banned, but there are still strict limits on what we're allowed to say. We cannot, for instance, have an actual discussion about why the basilisk wouldn't work.

Furthermore, there are aspects other than the ban that make LW look bad. Just the fact that people fall for the basilisk makes LW look bad all by itself. You could argue that the people who fall for the basilisk are mentally unstable, but having too many mentally unstable people or being too willing to limit normal people for the sake of mentally unstable people makes us look bad too. Ultimately, the problem is that "looking bad" happens because there are aspects of LW that people consider to be bad. It's not just a public relations problem--the basilisk demonstrates a lack of rationality on LW and the only way to fix the bad perception is to fix the lack of rationality.