This idea comes from the Catholic idea and interpretation of James 2:24 which says:

"You see that a person is justified by works and not by faith alone."

A Catholic would interpret this as they are saved by works and faith, works meaning acts in God's name and also faith in God himself. But the difference here with the protestant interpretation is that protestants look at it as "Sola Fide" which basically means just faith. So the protestant argument of this scripture is that we are saved through faith alone and different denominations will say different things about this but mostly it's that works contribute to your faith but they do not guarantee your salvation or contribute to it, just the faith that Christ died on a cross for us. Whereas the Catholics believe works directly influence your salvation.

I'm a Protestant myself but I'm an ACNA/ANIC Anglican and 90% Catholic pretty much in terms of my personal beliefs and agreeance with their theology. I agree with the Catholic perspective on this as to me it's more theologically sound and well researched and often the Bible is far less complicated than many scholars try to make it and Catholics often take the scriptures LITERALLY as they are written.

It's a bit heavy but if you're interested in diving deeper here's an article from "Catholic Answers" on the subject and they also dive into the protestant perspectives:

https://www.catholic.com/magazine/print-edition/faith-and-works-0

https://www.catholic.com/magazine/print-edition/arent-we-saved-by-faith-alone

Sorry I didn't touch on your question as exactly addressed because I needed to explain the distinction between sola fide and the Catholic perspective (I forget the Latin word) but in this case the atheists are partially right but you also need to have faith in God and do those works to glorify the Kingdom of God and his name (Catholic view). As pertaining to your question as to why atheists believe this, probably because they hear a lot of protestants make the argument that Catholicism is a works based denomination specifically. As for my opinion as to whether Baptist's are more on point, firstly, Baptists are Christians and are also under the Protestant category (anyone whose a Christian but not specifically a Catholic or Orthodox Christian), I told you earlier in this that I personally side with the Catholic viewpoint but I think it's interpretation is really up to the individual to decide.

Also this is a wonderful question to pose to a priest or an RCIA teacher that is knowledgeable. Honestly when I took the program this was the kind of stuff we addressed and I took it because I realized I was interested in theology and Catholicism has a deep, and very rich theology that is also quite logical. Unfortunately for me the explanation as to why I never became a Catholic is quite intellectually and emotionally deep but suffice to say I'm not 100% on board with everything Rome does.

I hope that answer was sufficient. People don't normally pick my brain on this stuff so I wasn't entirely perpared. Feel free to DM me with further questions. Although I'm nowhere near as qualified as a priest or scholar, so you'll have to accept my layperson answers.