So today after work I double-checked and actually, the fan is working! (Of course it was not running when the engine was shut down, as I had wrongly assumed after having seen several cars that have an electric fan that keeps on running even after the engine is shut down, if temperature is still high). Apologies for that.
Also, the auxiliary fan seemed to work just fine after I turned on the AC. Both fans running did seem to keep the temperature in control a little bit better, but when going uphill or in traffic, the gauge would still hit the third mark, although it did go back down to normal when going downhill.
So I'm ruling out the fans. However, I did notice some loss of coolant -- I checked it yesterday and filled it up to the mark, and today I had to refill again, quite a bit actually. Could this be a result of the overheating, or does it mean I have a leak there? Just in case, I checked the oil but couldn't see any signs of coolant in it.
My best guess right now is the waterpump, my last car (Toyota) had the same problem, and that's what it was there. I guess I'll just have to get it checked out at the garage ASAP.