From what I understand, the issue was building the station itself, but getting the laser to work, which they were so desperate to get Galen Erso to complete the project, because he was the only person that really understood the kyber crystals. So the Death Star could have just been sitting around for a decade or even two until the laser dish and some internal stuff were installed once the science was figured out. Though none of that is very overtly stated in the films.
And like you said, Nev, the second Death Star is actually three times larger than the first (which I personally don’t like, but at least you can’t really tell just from watching the movie), so you’d imagine it would take even longer than the first, but I think the lore spins it as it being easier to build since it is not the first. It seems in general building stuff in the Star Wars universe can be done rather quickly, but production scale is never really shown on-screen.
According to Wookiepedia, the DSII was project was started because of DSI’s destruction, and in Legends it started construction around the time of the Battle of Yavin as well.
Also, I believe the new canon (not sure about Legends) implies that the DS1 was not the first Death Star. Instead, the Death Star we see at the end of ROTS was the first prototype. I guess that is meant to explain the different positioning with the dish. But, if I recall correctly, that prototype ended after the Geonosians revolted during its production. I think this was all covered in the Rogue One tie-in novel Catalyst.
Honestly, I think any idea can be spun to fit into some kind of logical framework. My only major gripe against the single-Death Star idea is that it probably would be easier for the Empire to just build a new Death Star rather than repair a heavily damaged one. Like I’ve said before, it can analogous to a car. If you total a car, it would be easier (and possibly cheaper) to buy a new one rather than to try and restore the total one.
But, having one Death Star just makes that whole plotline a little more streamlined.