So, this thread has gone on several tangents now, but I'm still curious about the TS's original question:

Assuming the circuit designer had a choice, why would he/she use curves everywhere on a circuit that doesn't deal in super high frequencies?

 

After reading all the comments, I may do a silly board with curly tracks everywhere just to confused the hell out of someone/ just to have some fun

 

I used to draw PCB traces with sharp corners in the old days... until I noticed that the copper started peeling off at those sharp edges, and were also more prone corrosion over time ... those problems disappeared completely after I started drawing traces with a radius of at least 0.031" at the corners.

AFAIK: sharp corners also trapped etchant as the board is lifted out of the tank. Corners also focus magnetic fields and increase the chances of crosstalk.

 

I know that in the early days, there was a concern about corners (90 degree inside) could trap etchant which was harder to wash out. I think that issue is long past, though. [edit: dang, Ian beat me by 4 minutes]

My guess is that it's about aesthetics more than anything else. A fair number of designers try to throw some artistry into the process. Personally, I like my boards to look good - neat and tidy, things lining up, symmetries and such. I think of it as making the board look architectural though that's kind of "wine tasting sensibilities". The cynic might say "if you can't make it good, at least make it look good".