So other than propaganda, why did the Nazis build it?
I'm going to narrow this down to why did the Nazi government invest in a nationwide videotelephony service because discussing whether each person/company/organization was "the Nazis" is another question. Also because it's easier to trace to motivations of building a system requiring a large investment than a few people inventing a cool prototype FÜR WISSENSCHAFT!
I'm sure people came up with all sorts of ways it could be used that read like a sci-fi novel, same as how the Internet was supposed to change our lives in 1993, but found out the hard way it was too soon and too impractical. But propaganda value is the real reason. The Nazi govt was all about showing the world Germany had recovered from the (in the Nazi view) humiliation of WWI, that it was a world power again, and that Fascism was the better way. The 1936 Olympic games in Munich was televised.
It's a major achievement, but was it actually worthwhile compared to an equal investment in phone lines or other forms of telecommunication?
No. Like many inventions before and during WWII, it was too early. Too expensive and too impractical with the immature technology. And once the country went on war footing, they needed the money to blow things up.
Things would advance quickly and ten years after the network was shut down we'd have ubiquitous TV sets. Ironically, AT&T was predicting ubiquitous video phones in 1993 and they were still decades early.
Secondly, did anyone pick up on the research and technology for this service, which was revolutionary at the time?
I don't know what was done with the network, but the Germans developed a number of guided weapons and experimented with television guidance. The Henschel Hs 117 and Hs 293D were both TV guided, but fortunately the war ended before they could be perfected and put into production.
An interesting development of the missile system was the Hs 293 D . It was a standard HS 293 A1 missile with an extended nose and fuselage center, build to carry Television equipment, thereby removing the need for the carrier plane to keep line of sight to the target. In the rear section of the missile was an antenna sending signals to the carrier plane. The television equipment, K11, was crude compared to todays guided weapons, 224 line pictures been screened at a frequency of 50 pictures a second. Test flight were carried out in 1944, but the development program never really took off. The TV guiding by Joystick required quite some skill from the operator, and for the program to work there needed to be a large amount of simulator training. Range for the TV signal was about 4 km.