Yeah I laughed a bit when I saw Rudin’s book on that list. Understanding Analysis would be a better choice, even the end chapters of Spivak’s Calculus are a great into to Real Analysis.
I agree and I don't know why the myth persists that [baby] Rudin is the best way to learn analysis. It's certainly not so for a lot of people who could learn analysis perfectly well from friendlier (but not necessarily any less rigorous) sources. Rudin requires a lot of self-motivation.
In addition to Abbott which you mentioned,
"How to Think About Analysis" (Alcock), good gentle introduction for anyone who tries to dive into an analysis textbook and hits concrete.
"Real Mathematical Analysis" (Pugh), good, rigorous approach that doesn't require quite as much effort to power through as Rudin does, although some of the exercises are tough.
"Analysis I" and II (Tao), is good and starts from scratch building numbers from the Peano axioms, so there are no other unproven assumptions underlying everything later.
"Counterexamples in Analysis" (Gelbalm & Olmstead), very helpful for understanding some pathological cases that break theorems one might intuitively believe are true.
There's no reason today why someone can't grab pdfs of all of the above (even Rudin, although it's never been digitally typeset), from any friendly internet library, and use them all to build a better understanding rather than relying on one alone.
