Hatcher Algebraic Topology: I have all the prereqs, so why is this book unreadable for me?

Before, I scoured stack exchange for the prerequisites to read this book:

(See: Topology Prerequisites for Algebraic Topology, Module theory for chapters 1-3 of Hatcher Algebraic Topology, Learning Roadmap for Algebraic Topology, Algebra prerequisites for Hatcher's Algebraic Topology)

I got 100 on my point set topology course and am very comfortable with groups, rings, and modules so I thought I should be able to comfortably start learning out of Hatcher. I've heard great things about this book - how it's by far the most readable book in introductory algebraic topology, beautiful typesetting, builds up from basics, assumes little of the reader, etc.

But a couple pages in and I'm completely lost!

Hatcher keeps talking about orientable surfaces and genus - both concepts which are not once mentioned anywhere in Munkres, or indeed, in the majority of introductory topology courses. Neither of these concepts are defined in Hatcher either. So I thought - maybe I'll learn these concepts first and come back to the book! But no - every reference on genus I've found is another book on algebraic topology, one that needs prereqs beyond the ones demanded by Hatcher. All mentions of orientable surfaces lead me to differential geometry references - but I know 0 things about differential geometry.

The kicker is the prof I talked to who is running the course based on this book said I'll be fine in the course given my background knowledge. But from what I've skimmed from the book, I'm sure to fail the course at this rate.

I'll give a different example. The first time Hatcher defines the real projective space is here:

But is it just me or is this 'proof' extremely unrigorous and handwavy? I feel like half this proof relies on the reader's intuition about $\mathbb{R}^n$ for $n<=3$ and somehow this translates to general $\mathbb{R}^n$. The other half is massive leaps in logic that took me a long time just to think of a proof sketch. And this was just one example from the book, mind you.

Honestly, I'm thinking I've just hit a wall in my limits in pure math. What am I doing wrong? What makes this text readable to others but not me?

Or rather, my question should be - how do new people read this book in such a way that they don't get hung up in the way I am doing right now?

I highly recommend that you do not start with chapter 0, and if you really want to read Hatcher, just start with chapter 1. Chapter 0 is supposed to be extremely informal in spirit and can be skipped (he says this in the first para), and so it isn't meant to be scrutinized in that way. You are absolutely NOT hitting your limits in pure math; please don't be discouraged. I think a more gentle introduction to algebraic topology is Massey's "Algebraic Topology, an Introduction." It doesn't cover homology or cohomology, but it does the fundamental group very well. There are nice pictures in the book and it is a good continuation from point-set. Then you can pick up Hatcher at chapter 2 and start with homology.