Re "being paid", consider that he took early retirement in 1992 to work on this book that he sees as his life's work (https://cs.stanford.edu/~knuth/retd.html) (and until then was doing in his "side" time, separate from his job of teaching and research), giving up whatever salary a tenured Stanford CS professor's would be, and has been living off savings and royalties since then: he only gets "paid" when people choose to buy the results. (And if he were to focus on money he'd probably put out more newer volumes than 2 in 30 years, rather than keep making newer editions / fixes to earlier volumes…)
But you're right that Knuth has a dream "job" that the rest of us, at least those with the slightest "scholarly" inclinations, can only envy him for: he reads, explores, digests, and explains, the topics he likes and in the way he likes, and the results are great. But when you're already a legendary programmer in college (http://ed-thelen.org/comp-hist/B5000-AlgolRWaychoff.html#7), and have won a Turing Award at 36 (http://hagiograffiti.blogspot.com/2009/01/when-will-singular...) for a book you've been writing in your free time, one can say you're entitled to the life.
You can get a bit of an idea by reading Knuth's "The IBM 650: An Appreciation from the Field", written in 1986 and nostalgically looking back at the 1956–60 period when he entered college and first encountered a computer (the IBM 650). The paper is available here and is gripping reading: http://ed-thelen.org/comp-hist/KnuthIBM650Appreciation.pdf
I consider this the first and last generation of truly self-taught programmers. Earlier programmers either built the computers themselves or were hired and trained on the job; they were professionals. Later programmers (all of us) did have some programming books/tutorials or some sort of community to draw from. But Knuth and his peers just had a machine with a manual describing its opcodes, and practically "discovered" programming independently at each "site", e.g. college. As he writes:
> The IBM 650 was the first computer to be manufactured in really large quantities. Therefore the number of people in the world who knew about programming increased by an order of magnitude. Most of the world's programmers at that particular time knew only about the 650, and were unaware of the already extensive history of computer developments in other countries and on other machines. We can still see this phenomenon occurring today, as the number of programmers continues to grow rapidly.
But you're right that Knuth has a dream "job" that the rest of us, at least those with the slightest "scholarly" inclinations, can only envy him for: he reads, explores, digests, and explains, the topics he likes and in the way he likes, and the results are great. But when you're already a legendary programmer in college (http://ed-thelen.org/comp-hist/B5000-AlgolRWaychoff.html#7), and have won a Turing Award at 36 (http://hagiograffiti.blogspot.com/2009/01/when-will-singular...) for a book you've been writing in your free time, one can say you're entitled to the life.