Anatta/Anatman viewed as no existing self or soul is one of the biggest misconceptions of Buddhism. And, the same goes for the view that Buddha believed that the "self" did not exist.
Part of the problem is that the Buddha did try to get his monks to a state of selflessness.
What is Anatta trying to explain:
- all things are an aggregation, even those things we think are an essence; and
- all things are influenced by causality, even those things we think are an essence.
What does this mean: things are complicated, they will have several component parts, and each part is subject to causality.
Now, What Buddha has been trying to teach his monks was to evolve to the point where "YOU" are Selfless in your thoughts actions, and speech. So, after you have found "your-self", lose "your-self" in all the good you do.
So just do good as "selfless acts", these acts will bring "you" closer and closer to nirvana, a state of total selflessness - in "your" thoughts, actions, and speech.
Oh yes, you do still exist, just not driven by selfishness, to the extent that "you" will not even be driven by self-interest.
And yes, on a gross level, others see you as selfless; but at the subtle levels only you can know when self-centeredness arises in you.