Actually:

• He had already published the first editions of Volume 1, 2, 3, and the second edition of Volume 1, by 1973. It was in 1977 when the publishers sent him galley proofs for the second edition of Volume 2, having switched to phototypesetting (away from hot-metal typesetting a la Linotype, though IIRC it was actually Monotype) that he was disappointed with the results. And he had some back-and-forth with them and they did improve their fonts (https://tex.stackexchange.com/a/367133/48), but he was still dissatisfied.

> I didn't know what to do. I had spent 15 years writing those books, but if they were going to look awful I didn't want to write any more.

• At this time he came to know of the existence of digital typesetters. Typesetting with computers had existed before, but it had always seemed a crude toy, rather than something suitable for “real books”. But he saw Patrick Winston's Artificial Intelligence that had been just published (I think he got an early proof copy to review or something), and he realized for the first time that digital typesetting was an option (apparently Winston's book was printed at >1000dpi, and Knuth later got his hands on a machine that claimed a resolution of 5333 dpi: see this wonderful comment from Knuth's student and “right-hand man”, David Fuchs: https://news.ycombinator.com/item?id=20009875)

• In fact it was the fonts that he was dissatisfied with rather than the typesetting, so METAFONT was in some sense the primary/motivating project and TeX was only written in order to be able to use METAFONT.

• Actually his first idea was to simply take the old fonts, get high-resolution scans of them (not easy to obtain at that time) and use them directly. He approached Xerox Research Center but:

> I asked if I could use Xerox's lab facilities to create my fonts. The answer was yes, but there was a catch: Xerox insisted on all rights to the use of any fonts that I developed with their equipment. Of course that was their privilege, but such a deal was unacceptable to me: A mathematical formula should never be "owned" by anybody! Mathematics belongs to God.

• So he went home and (after trying a bit with TV cameras) tried projecting photographs of the pages onto the wall and tracing the outlines, and it was while staring at these images that he realized that the shapes of letters were not arbitrary but there was some logic to them (e.g. in the font he was using, the spacing between the vertical strokes in 'm' was equal, and equal to that in 'n'), and he decided (as a computer programmer) to capture this design in code — something that had never before been done. The hardest letter to capture this way is S, hence the paper in the OP.

> Finally, a simple thought struck me. Those letters were designed by people. If I could understand what those people had in their minds when they were drawing the letters, then I could program a computer to carry out the same ideas. Instead of merely copying the form of the letters my new goal was therefore to copy the intelligence underlying that form. I decided to learn what type designers knew, and to teach that knowledge to a computer.

• This is also why METAFONT never really caught on among typographers: as Charles Bigelow (quoted by Richard Southall, https://luc.devroye.org/Southall-METAFONT1986.pdf) observed, “the designer thinks with images, not about images”. Knuth did not want crude “geometric” constructions of letters (as some prior 16th century typographers had attempted: https://www.ams.org/journals/bull/1979-01-02/S0273-0979-1979... and as some typographers only passingly familiar with METAFONT think!). He wanted actual real typographically beautiful shapes, but to be able to generate those shapes with code. This is obviously much harder than simply drawing the shapes using visual intuition, even if it enables variation. (See “The Concept of a Meta-Font”: https://gwern.net/doc/design/typography/1982-knuth.pdf — again, many people in the typography world confuse the abstract concept of a meta-font introduced in this paper with (their incorrect impressions of) the METAFONT program, and omit crediting Knuth for variable fonts).

• The second edition of Volume 2 was not printed with Linotype. Yes the machines still existed in Europe and he talked to typesetters (he mentions in particular a person from Belfast), but it was in fact published using TeX (the first version, TeX77 and MF78). He was still unhappy with the results, though, and spent a few more years learning more about typography and working with people like Bigelow and Hermann Zapf, before the rewrite into the current TeX82 and MF84 (and current version of Computer Modern). I think it's only with the third edition (1997) that he's finally satisfied.

The Letter S, by Donald Knuth (1980) [pdf]