I disagree on this point - I'm pretty sure all the big 3rd party apps at least have ads. The problem is fundamentally just that the 3rd party apps are a lot better than the official app, and have been for some time. If Reddit had made the official app better (which they've had _years_ to do) then significantly less people would care about any of this.
They also could have gone the Spotify route, which I think would have gone over significantly better - Keep the API as-is, but require a paid premium account login to use it. Functionally it's not even really a difference, but it means Reddit deals with all the details rather than the 3rd party apps. However, functionally the goal was to simply price the 3rd party apps out of existence, so that's probably why they didn't do this.
It's also pretty clear from the response that they never thought this through, which is hard to believe. They had to have it pointed out to them that tons of stuff currently uses the API which has no replacement, you'd think they'd have reviewed what currently uses the API before drastically changing it. Reddit has gotten significant value for free by having people write code against their API, that's code they didn't need to write themselves.
I use(d) Relay for Redit, paid a small ($5?, maybe) one time fee for the "Premium" version a few years ago and have never see an ad. You are correct in the sense that if you use the free version of Relay (and possibly others), you would have ads from the app, not Reddit, but if you're willing to pay a small amount, you can get rid of them.