Jay Cummings' Real Analysis. Cover a little less than a year's worth of real analysis at the university level and can be used for self-studying (I used it myself to stay a few weeks ahead of what we were currently doing in class and it worked out great). It explains stuff intuitively, gives specific "scratch work" sections prior to proofs etc. while not sacrificing any rigour.

>a lot of proofs are a little idiosyncratic and more complex than they need to be

Do you have specific example of one of those? I don't have the book so I can't look the one you mentioned up. It's certainly possible, but rather unlikely and given that you're just getting started with the topic and likely don't have a ton of experience with proofs it's possible that your simplifications actually break the proof. Maybe it's also really a simple idea but expressed a bit convoluted or smth like that - some authors just aren't good writers.

>Will I eventually get better at learning solely from textbooks?

I think this depends a lot on you and how you use the textbooks - some people are very good at self learning and benefit a lot from it, others aren't. It also depends on "mathematically mature" you are already I think.

>And is this just the normal self-study experience when it comes to undergraduate and graduate mathematics?

Often times: yes. A lot of textbooks are terrible for self study (at least at the lower undergrad level). On your own, without any guidance or people checking your work etc. it can be very hard (compared to when you're taking an actual course on the subject) to develop maturity in some subject.