I don't think you should conflate free price and free source. [A]GPL is about availability of source code. It's end-user protection. End users should be able to know what's going on with their software and should be able to tinker as they see fit. You can sell your software for money and have stipulations that the source is always made available and that changes are made public. Non-paying entities simply do not have the legal right to use your software, they don't have license to use it. When you include a license with your software, you're allowing people to use it (under those terms). If you don't give/sell them a license, they don't have right to use it.

I think a high visibility usecase of AGPL would be Redis vs Amazon. Amazon took the, then more permissively, open-sourced Redis and basically "selfishly" tweaked and altered it without contributing back upstream, and then began selling that as a service. Basically taking someone elses thing and selling it as their own (which was well within their right with the MIT or whatever license Redis was using at the time). So Redis re-licensed as AGPL.

You could also release it under more than one license. Offer a FOSS license generally, but offer to sell proprietary licenses for those that don't want to disclose their source/otherwise follow the terms.

FSF actually mentions monetization in one of their articles, I think, How to Choose a License. Worth a read. [A/L]GPL are definitely not adverse to monetization, it's more for end-user protection.