If you have a hypothetical continuum of charges moving in a circle, and you use Maxwell’s equations, you’ll find that given a constant charge density, no EM radiation will be emitted.
If you have a superconductor, you should really be using quantum mechanics to understand it. If you can imagine that an electron can “orbit” a nucleus without emitting EM radiation, then you can imagine that current can flow in a loop without emitting EM radiation. The behavior cannot be explained by thinking of the behavior of a single electron, but must be explained by considering the behavior of many electrons in a quantum mechanical system.
Also note that the actual speed of electrons (the “drift velocity”) moving around a loop of wire is extremely low, so if you treat electrons as point charges and ignore quantum mechanics, and you calculate the amount of EM radiation that should be emitted by a typical loop of wire, you will get an extremely low amount EM radiation emitted, which would be very difficult to measure.
Those crazy electromagnets they use on these stellarator are simple superconducting loops that they 'charge' by inducing a current. That current is maintained so long as the superconductor stays below a certain temp. There is even something called a SMES (superconducting magnetic energy storage) that stores power this way, as I understand it they have a 0% self discharge rate.