For professional applications there are sulfur plasma lamps which have a continuous spectrum at high efficiency. Unfortunately they aren't economical below about 1000 watts which is impractical for many applications.

The technology basically works by continuously microwaving (think oven) a small amount of sulfur gas. The development of solid-state microwave emitters — most microwave generation is still done with vacuum tubes — might help miniaturize the devices. However, it's hard to beat the simplicity of an LED.

There is, but most humans are used to the spectral pattern of black body radiators at all color temperatures. Be that sunlight at higher temperatues or fires / candlelight at lower temperatures.