You do, by the way, see such circuits pretty often -- they're not impossible to start up. Usually what's done is a combination of very generous soft-starting (so the massive surge currents are brief at first), very low resistance switches (usually a big pile of IRFZ46Ns, for automotive (12V) input), and excessive leakage inductance in the transformer (which is usually a toroid, wound in no particularly special way).
Which, putting those clues together, yep -- that's almost every automotive power amplifier out there. A DC-DC converter, turning 12V into +/-30V say, followed by either a class AB (linear) amplifier, or class D (switching) amplifier. The 30V rails are unregulated -- the converter runs at full duty cycle (typically a TL494 or similar, so 45-48% duty per switch), and the supplies are proportional to the input.
They're also notorious for exploding. Dead power supplies are probably the most common failure of those amps.
The better quality ones do actually have current sensing and a filter choke, and almost never fail -- they control one of the two things that destroys silicon, so it's never allowed to reach an explosive value in the first place. Proper design approach. (The remaining factor is excess voltage, which is a bit harder to deal with, but also not insurmountable.)
Tim