Software that isn't executed on a user's machine is out of the user's control. The GNU AGPL redefines distribution to include serving over a network so that software served over a network must disclose source code as well (unminified JS, server-side code, etc). It's also copyleft, which is good: the purpose of permissive licenses is to enable proprietary derivatives, which is against the spirit of the Fediverse.

One interesting consequence of Mastodon's AGPL licensing is that Gab, the right-wing twitter clone that runs a Mastodon fork, also has to disclose its source. That was how people found a vulnerability in Gab that allowed the recent "hack".

As a service operator, why would I want to choose software licensed this way?

Because what you're offering is a service, and not someone else's software. Your added value is not actually in the code or the cute pictures; it's being available, having a low latency, allowing people with no tech knowledge to take part in the discussion, building a relation based on mutual trust rathet than serfdom. It's fundamentally the difference between a user and a consumer.

As an operator you're always making a tradeoff between the "negatives" of a license and the "positives" enabled by the actual software.

So for one example, the recent (bait and) switch of Elasticsearch to SSPL effectively means that for many actual real-world users of Elasticsearch, the benefits of staying on the official Elastic branch has ceased to be worth the costs of the license when your alternative is running FOSS Elasticsearch/Opensearch.

So to answer your actual question - presumably as an operator you want to use the AGPL-licensed code more than you want to avoid those same negatives of AGPL.

That's exactly why I license everything I open-source MIT/Apache.

I find that issue very confusing. How is a GPL-2 license not acceptable for being listed on a website of GNU projects?

I note they went along with the relicense but the site only currently lists "GNU FM" and "GNU Social". Is there an actual compelling argument for a project to choose a license just to be listed alongside those?

GNU FM and GNU social are both AGPL projects. The gnu.io site was for GNU projects that were AGPL.