Very strange things are going on. I rewired almost everything with shorter wires like you suggested (I was using cheap male-to-male wire connectors and switched to stripping solid core 22 gauge wire) and this time I did nothing to pullup SCL, like before. This time the output seemed to be working, but displayed the time as 3:50, which is approximately the time I tried pulling SCL up with the same resistor as SDA. So I tried pulling SCL up again, and just like the last time I pulled it up, nothing worked and nothing output to the serial monitor. Just afterwards I tried the circuit without pulling up SCL and this time it reads 4:39. The actual time was 4:41, and 4:39 is when I tried to pull up SCL.
So I did this procedure over again, except tried to switch it as fast as possible, between having a pullup resistor and having no pullup resistor for SCL. I found the quicker I can switch it, the more accurate the time displays. So for some reason the chip's time is set when SCL is pulled up, but can only display the time when there's no voltage?
Just took some pictures, they'll up in a couple minutes.