Do I believe Atheists are going to hell?

[Neo-Scholastic]

Does feeling remorse for wrongdoings, acknowledging that you are not perfect, and wanting to be better, in and of itself, count as some sort of recognition of the need for redemption?

Once the need is recognized, sure, that would be a step in the right direction. A person still needs to look outside themselves for both salvation and forgiveness. As the Schoolmen said, "you cannot give what you do not have." That is why atheists cannot go to Heaven. They have arrogated unto themselves a power they do not have - the ability to absolve sin.