Why don't loop currents produce light?

Circular currents do produce EM, and indeed this is exactly how X-rays are produced by synchotrons such as the (sadly now defunct) synchotron radiation source at Daresbury. In this case the current is flowing in a vacuum not in a wire, but the principle is the same.

Current flowing in loops of wire don't produce radiation in everyday life because the acceleration is so small. The electrons are moving at the drift velocity, which is only around a metre per second, so the amount of radiation released is immeasurably small. Synchotrons produce radiation because the electrons are moving at almost the speed of light.

If you consider that electric current is actually the flow of individual charged electrons, then as John Rennie pointed out, the radiation exists but is negligibly small. But if you were to imagine breaking the current into more and more point particles with less and less charge while holding the linear charge density $\lambda$ fixed, then the radiation would decrease more and more, because the radiation per particle decreases as $q^2$ while the number of charges only increases as $1/q$. So in the actual continuum limit where the current becomes perfectly steady, the radiation actually vanishes completely.

To back up John Rennie's answer, consider the Bremsstrahlung formula for velocity perpendicular to acceleration: $P= {{q^2a^2\gamma^4}\over{6\pi\epsilon_0c^3}}$. For all practical purposes $\gamma=1$, so we can simplify this to $P\approx ({q \over \mathrm{C}})^2 ({a\over \mathrm{m/s^2}})^2 {1\over{18.85\times 8.85\times 10^{-12}\times 2.7\times 10^{25}}}\mathrm{W}\approx ({q \over \mathrm{C}})^2 ({a\over \mathrm{m/s^2}})^2\times2.22\times 10^{-16}\mathrm{W}$.