This pushes the limits of my section 230 knowledge, but I think you've got this backwards. A company that wants to comply here needs Section 230 to exist, a company that is okay with ignoring Google doesn't care about Section 230.
Section 230 immunity isn't necessary for things that are completely unmoderated. If the comment sections are literally entirely unmoderated, they fall under the pre-existing statute (Cubby, Inc. v. CompuServe, Inc).
However, if the site wishes to moderate comments for some reason, they could be held liable for comments that stay up but are problematic (libelous etc.). So without section 230, a site would be in a catch-22. Section 230 continuing to exist avoids this problem.
You have this exactly right. It's a fairly straightforward law. It's kind of bizarre how far people (especially journalists!) misunderstand it.
This ignores the factual reality that Google is a monopoly in several verticals, and the number of companies that can ignore Google is actually zero. Every company needs to be in Google's good graces whether it be for advertising or app installation on mobile phones or search visibility.
Even all of Google's direct competition in any given market needs to support Google in other markets. Google is inescapable and compliance with their policies is as mandatory as actual law.
Google should absolutely have the right to say no to running ads on sites they think are objectionable. The issue is that there's not a healthy market.
Honestly it would be better for this sort of thing to be banned from the top down by governments, but they seem loath to call any "white" nationalist group terrorists no matter how many weapons they bring to rally and how many people they injure.
White is in scare quotes, because you never know who they are counting aa such...