According to some moderators the official moderation tools are not good enough to handle large subreddits so many moderators use the third party tools.
Most moderators overwhelmingly rely on the utility function you get in 3rd party apps that isn't currently in the official app, so I genuinely believe this is one aspect of the protest. I could be wrong, but I genuinely can't see a situation where this amount of action is purely around seeing ads.
Pretty unanimously, 3rd party app developers have said they'd happily show Reddit's ads in their feed, but Reddit never came out with this feature. I don't think any 3rd party app developer was expecting a free ride on Reddit's API forever.
Official numbers show that only 3% of moderation actions are performed via third-party apps, putting a definite nail in the coffin of that fabricated argument.
I think from doing extremely light research, it seems pretty much _all_ the 3rd party apps are better for moderation than the default client. However it seems unanimously people tend to prefer to moderate on desktop.