That picture of the telescope is from Xena, warrior princess.
Regardless, tube or lenses, it looks to me like at least one part of the telescope is missing, maybe more.
Assuming we get the telescope working, if it's a regular passive star-viewing telescope, I suspect we'll be looking for the relevant guardians' constellations. Maybe the constellations on the TCG art, or perhas the star patterns from the ceiling panels. Maybe finding the constellations will reveal something to an astronomer somewhere and the plot will proceed.
As an aside about the ceiling panels, in spite of multiple attempts to line the stars up with the images of the statues and/or the TCG art, I don't think they're supposed to be representations of the constellations. All the gems are the same size, none are brighter or dimmer or different colored. Still they seem to be part of the puzzle. Maybe they'll change a bit as the plot progresses and as the file names seem to insinuate.
Now if the telescope is some kind of magical statue-animating faerie-releasing historic telescope-of-epic-proportions integral to the plot, there might be some kind of game or riddle or thing to make it work. Heck there migth even be a quest to get the telescope parts and then do whatever needs to be done.
Maybe there will be another NPC introduced related to the telescope. That would be cool.