interesting. huge photo, but out of focus.
the 4 diodes are in a bridge arrangement, so we might assume this board has DC.
the clock/display board has 4 Tact switches, which further promotes that the board has DC.
if the clock has/is a micro, then there is a possibility you can replace that with an on/off signal.
the heater has two wires. the red one goes to a bi-metal switch. that is the entire temperature control circuit. too cold, the switch clicks for heat. too hot and it shuts off.
the large yellow block is most likely a cap to prevent a huge surge when the bi-metal control cycles.
if you want to get very hot water, just replace the timer with the arduino. nothing else needs to be done.
if you want to gut the thing, keep the bi-metal switch and put a temperature sensor on the coil and replace the existing relay with a SSR and then pulse the heat.
I always recommend opto's to isolate power and signals.
btw, you may be able to buy different value switches, so you could alter the temperature control.
as for temperature control, you are getting BTU's not temperature. faster water = lower temperature, but the BTU's will be the same as slower water and higher temperatures. as long as the heater is working. at least withing a window of efficient of the heater.
bottom line is that you will have a mains power unit. need only 1 power outlet, and can power both this and a voltage regulator or switching power supply for low voltage.