I will sound harsh, but all of this is true. Sorry, but both your circuit and control algorithm is a piece of junk. I wouldn't touch with a barge-pole something clicking all of the time and with slow, loose and dumb control algoritm. Big delay before reacting to temperature drop or overshoot and no proportional heating control according to actual temperature drop. 10s of heating while not touching some massive solder joint will result in huge temperature overshoot. It might be somewhat fine with bulky tip/heater which has slow reaction to heating and heat sinking but with cartridge type tip/heater this is unacceptable.
As the result, you'll be far better with $30 FX-951 knockoff (BAKON 950D), even if using knockoff cartridges. And can be improved by using genuine cartridges and handle.