Element power can be limited either by controlling the effective voltage using a phase-angle controller (i.e. dimmer) as mentioned above, or by switching elements between full-on and off. In your application, there is a definite advantage to full-on / off method compared to phase-angle control. This is because what you are trying to achieve is a definite maximum currrent, limited by the 60A supply, but within that limit you want to obtain as much power as possible. It follows therefore that you must aim to get as much heating power per amp as possible, which means having the highest possible load power factor, preferably unity.
With a resistive heating element full on, the power factor is unity, but as soon as you start reducing the power by phase-angle control, the power factor starts to drop due to the heavily distorted current waveform. So although you can use this method to keep the current within the 60A limit, you will be unnecessarily curtailing the power input to the thermal store due to the lowered power factor. In short, with phase-angle control, at half the amps, you will have less than half the watts. Whereas if you switch on half the elements, you will have half the amps and half the watts, which is what you want. Industrially, the preferred way to regulate heating power is by burst firing, repeatedly switching the element on and off for a few cycles. The problem when the element load makes up much of your total, is that each switching cycle is likely to cause lights to flicker etc due to fluctuations in the voltage drop of your supply.
Probably the best and simplest method as mentioned above is to switch the heating in steps of one element at a time, based on load thresholds, and the device to do this is a 'load shedding controller' or 'load shedding relay' which is what you need to be googling for. The main circuit for house load passes through its sensing circuit, while its output contacts operate a relay to disconnect the element when the house load exceeds the threshold set on the dial at the front. You could put three single-channel units in a chain with their thresholds set differently, each controlling one element. Here's a very simple example, I think this one only goes to 20A but it clearly shows the configuration:
Load shedding relay 2...20A ELAR-20 - https://www.eibabo.uk/eberle/load-shedding-relay-2...20a-elar-20-eb11101084
Obviously if the element load goes through the sensing circuit as well as the house load, the controller would need to have a hysteresis (dead band) greater than the element current so that it doesn't hunt. This is programmable on the more sophisticated units, which can also offer multiple channels.
I haven’t time today to delve into why they are wrong
4.2kJ/kg/K - correct.
Heat input for 30°C rise = 30 * 4.2k = 126kJ - correct
The error is in conversion from kJ to kWh
126/3600 = 0.035 kWh.
I am not sure about the temperature limits. With a thermal store one would definitely want to be using a higher maximum temp, e.g. 90° C. What the lower limit would be depends on the application. If you were considering heating water for domestic use using an amply-sized counter-flow exchanger, you could go as as low as 30°C and therefore the thermal store differential would be 60° storing 0.07kWh/l and requiring only 650 litres to store 45kWh. But 30° is too low as a practical return temperature for normal radiators so not all of this differential could be utilised for space heating. The return to the store from the heating circuit might be nearer 45-50°C.
