The dimmer could be a switched transistor dimmer circuit that changes pulse rate of the AC. that would work if it is power rated 700 watts continuous. The dimmer could also be a variable resistance that changes voltage--that is not so good because as the voltage goes down the amperage goes up (OHM's Law) and the heating coil would likely be looking at more current than it is rated for. This would shorten coil life substantially; a 700 watt variable resistor is bigger than the box shown in the picture also..
Heating coils are inductive resistors and behave the same way as any resister does, when the voltage goes down, the current goes up (OHM's LAW) and the resistor takes a beating trying to dissipate more current. A current limiting circuit would assist in that area but those are not used in dimmers.
Switching transistors to handle 700 watts are expensive and require a substantial heat sink mounting. The light dimmer in the picture is for home lighting and not rated for anywhere close to 700 watts. It might last a while but its future is dubious.
A speed controller for power tools rated for 700 watts would be fine and handle the duty but it may not handle the surge current turning on the heating element. Harbor Freight has one that is inexpensive, but risky.
The standard thermostats that come in melting pots are just thermostats, they are a variable timing circuit that turns the heating element on at full power for time intervals set by the rotary dial. That is why you can hear or feel the hum when the thermostat switches the coil on and later hear the silence when it turns the coil off at the interval you set the pot at with the rotary dial . The rotary dial itself is a low wattage variable resistor in a timing circuit for the main contact switch for the coil.
An electric hot water tank thermostat does the same thing turning on and off the heating element at full power timed intervals that you set.
A discreet device called a thermistor can also be used for this function and that actually senses temperature, but very impractical for melting pots.
PID controllers do work very well and can be home built if you are accostomed to populating circuit boards with components. They are not a complex circuit, kits are available on the web too, just watch the wattage and duty rating and get one that fits and a temp probe that will work with the kit.
Gary