Why doesn’t the comparator output go near the power supply voltage?
You did not specify what was the difference between the supply voltage and the output. According to the datasheet there could be as much as 5V difference for 2kOhm load. Basically you have only 330Ohm load - assuming that the LED voltage is negligible compared to the output voltage - , so the output voltage could be even lower. 


PS: next time please rotate the picture! It would be easier to read.