I read somewhere that Sturmey-Archer introduced the five speed hub in 1967, if my memory is right. I'm guessing that was in response to the 10 speeds, and the expense probably kept them from being more successful, but I don't really know. How long did they make them? There was a S. A. hub on EBay last year, and it looked like it was seven speeds, made in England. Did Sturmey Archer make a seven speed before they were bought out?
The 7-speed you saw was probably the late 1990s-era S7 hub. It's an interesting concept, but too little too late for Sturmey in England. By then, the manufacturing tolerances were not as good as in the immediate post-war years as well, and then the shifters were a more modern, plastic thing.
You raise a good point about the S5. The S5 went through a few variations, with some being pretty awful, and others pretty good. The earlier ones are good, but there are a couple later variations that had issues. The S5 has a bit of perplexing early history. The early S5 really is an FW four-speed with a bell crank added to the non-drive side in order to activate a combination of sun and planetary gears that give you very high overdrive. The FW lacks this because it lacks the bell crank, but it's feasible you could modify the FW to take a bell crank and then have, effectively, an early S5 hub. Sturmey should have done this earlier than it did, and then developed the idea a bit to incorporate a 'no neutral' or 'no in-between-gear' feature. The combination of 5-speeds and no neutral would be really nice to have in, say, 1955-60. But they didn't do it. They had instead continued producing the FW while trying to develop the cheaper SW hub, which was a disaster.
I do think the S5 is a somewhat late attempt to keep pace with 5 and 10-speed derailleur bikes:
https://bikeshedva.blogspot.com/2017/04/trying-to-keep-pace-with-derailleurs-4.html
It's not a bad idea, but it was impossible for Sturmey to keep pace with 10-speed (later 12-speed) systems, given the money they had to operate. I think Sturmey was, frankly, somewhat starved for money and pressed by its parent company by the mid-1960s to operate as cheaply as possible. Sturmey designs showed such promise from the early years right up to the 1950s, but then the SW marked a turning point. By the time of the S5, we're talking rather low-hanging fruit to squeeze one more, rather high, gear out of the FW design (which itself was 20 years old by then). Even that failed at first - the earliest S5s came with awful, low-quality plastic friction shifters that broke. The earliest S5s also have a weak, sheet metal bell crank that breaks. These had to be revised, and Sturmey did so within about 2-3 years of issuing the S5.
So I think the S5 Sprite represents the last in an evolutionary line of the "classic" Sports light roadster bikes. You have the Sports frame (a design firmly rooted in the 1930s-40s period), the heavy rear Prestube rack (another 1930s-40s type item), all-steel construction (getting dated by the 1960s), and the S5 hub. You're trying to eek more performance out of a very old school design. That's actually why it appeals to me - it's the final technological development of this particular line of bikes (though this general style of utility bike is still around, just with more modernized components and newer materials).
There was also an option with a 3speed hub with a twin or triple rear sprocket in conjunction with a rear derailleur. Giving 6 or 9 speeds, never tried one personally.
There was also a 3speed fixed gear hub, would love one of those!
It's known as 'hybrid' gearing and was particularly popular in the 1950s-60s era. The Cyclo line of conversions comes to mind. They're really neat, but you have to watch which gearing you use with the multiple cogs so you don't get too much overlap in your 6 or 9 speeds. But you can build a really sporty machine out of those.