My grasp of group theory, measure theory, and functional analysis are fairly weak, so maybe I'm not the best person to comment on this, but I think the problem may be that you were overoptimistic when you attempted to read those books. Usually when I read books on subjects I don't understand, I don't understand the book the first time I read it. Reading several different books on the subject helps. This requires a lot of persistence and tolerance for frustration. But that's true when you take a class, too!
As you say, though, you need to solve a lot of questions (which I interpret to mean "do a lot of exercises" or "do a lot of problem sets") to understand something. Reading a textbook without doing exercises is minimally useful, although it can help with the "roadmap"/"relationships" thing. Wikipedia is usually a pretty good roadmap, too, although it varies by field.
But you can also read textbooks and do exercises. This depends on the existence of, and access to, sufficient textbooks and exercises, but Library Genesis has recently extended that kind of access to most of the world. Taking functional analysis as your example, the 1978 edition of Kreyszig is on there, and it averages about two exercises per page, and has answers to the odd-numbered ones in the back. This quantity of exercises seems like it would probably be overkill if you were taking a class in functional analysis and could therefore visit the professor during office hours to clear up your doubts, but it seems like it would be ideal for self-study. And if two exercises per page isn't enough, you can get more exercises out of a different textbook, like Maddox (1970 edition on libgen) and Conway (first and second editions on libgen). You can find textbooks on scholar.google.com by searching for the names of general topics and then looking for "related articles" with thousands of citations, because for some reason people like to cite their textbooks.
Unless you can find a desperate adjunct math faculty member looking to make some extra bucks on the side or something, it's true that comparing your answers to the exercises to those given isn't as good as having a TA actually correct your homework. But it's usually good enough.
(Of course you should only download these books if that wouldn't be a violation of copyright, for example, if their authors granted libgen permission to redistribute them or you live in a country not party to the Berne Convention.)
Progress will be slow. But I think the key thing here is to start with low expectations: expect that you'll manage to read about 15 pages a week and understand half of them. I don't think you have to be a Terence-Tao-level genius.