> lengthen and shorten the light part of the circle according to your local sunrise and sunset times!

It's really not obvious, especially in light mode, but if you switch to dark mode you can see the day circle has a "bolder" and a "thinner" part. Bolder is the daylight hours, and its length and position corresponds to the daylight hours (in this instance, of Zagreb, since that's the primary timezone for the clock).

You're correct in that it can't work if you have multiple lattitudes. I took the easy way out - I ignore that and just use the first one :X