Most gentle introduction to undergraduate analysis

Take a look at Understanding Analysis (2nd ed.) by Stephen Abbott. The book is pretty well-known these days for a very clear exposition of most of the basics of real analysis. I think most of the topics you mention are covered, if not all of them. In common with Bartle it gives a kind-of 'basics of topology in $\mathbb R$' in one of the early chapters to make some of the later proofs a little smoother. A nice thing about it is that every chapter starts with motivational example/s to demonstrate to the student why the material is worth studying. The only drawback is that no solutions manual exists to my knowledge... but if you're using it for a class this may be a good thing!

I adore Spivak's Calculus. I think it covers every topic you mentioned in a completely rigorous fashion, and motivates the definitions (for example, he builds up to the definition of the limit until it seems like the only reasonable thing to do). It's also full of interesting problems.

Introduction to Real Analysis 4e Bartle. It is the perfect transition from calculus to analysis; I was able to self-study from it. It is gentle as you have required and covers every topic you have listed and more such as basic prelimiaries, a generalization of the Riemann Integral, and a basic introduction to topology.