Is it true that any system of accelerating charges will radiate?

The claim that accelerated charges must radiate is simply false. There are very many simple situations in which they do, but in general things should be examined on a case-by-case basis; there is not simple thumb rule like "acceleration yields radiation."

The simplest way to see this is to consider a wire carrying a constant current. This situation is magnetostatic, with well-understood electromagnetic field given by Biot-Savart law... which trivially lacks any radiation, as the field is constant. This is independent of the shape of the wire, and yet, if the wire is not straight, the charges within it must accelerate at some point.

A more exotic could be a uniformly rotating ring of charge. Both charge density and current density will be constant, and will produce constant electric and magnetic fields. This can be experimentally realized with toroidal superconductors.

I tried to think of a simple system of charges that would disproof this statement, since I intuitively think that it isn't true, but I haven't come up with any

The simplest system is an hydrogen atom composed of a proton and an electron. This system doesn't radiate if you accelerate it.