You are right, the final output dynamic range is limited by your output device, you can't circumvent that.
The 1,000,000:1 is the contrast ratio you might see on a sunny day, and represents the state of the world that your eyes are seeing.
Your eyes aren't capable of 1,000,000:1 contrast either, at least not globally across your field of vision, so we can play tricks in rendering to simulate what you would perceive with your limited eyeballs on a 1M:1 contrast ratio day, and this is what HDR rendering does.
Internally, the rendering code handles luminance well in excess of its ability to display on the final output device. All the game rendering happens internally in HDR, preserving this high level of contrast. The final steps of producing a displayable image are doing something to convert non-displayable "real" contrast ratios into post-processed simulation to fool you into perceiving high contrast scenes. This last part is subjective and artistic, since you are trying to figure out a way to show that high dynamic range scene. Common tricks played here are the light bloom from bright areas, the artificial amplification of darks areas, etc.
All these in-between rendering steps compose the result of multiple rendering passes into an image. Each pass which is blended with the results of a previous pass is subject to quantization error. The more passes you do (and modern games do lots), the more this quantization error accumulates, and leads to pretty ugly banding artifacts. You want to do your internal processing in the highest color resolution that you can, so that you only have to worry about the error of quantizing the 1M:1 scene into a 256:1 output device. You don't need the full 20 bits, and common practice is to use 16-bit S10E5 floats since hardware handles this quickly.
HDR televisions, while nicer than nothing, don't help you all that much. Displays have generally been 8-bit per channel, giving you 256 values distributed along a gamma curve which maps to a device with perhaps 3,000:1 contrast capability. HDR tv's give you 1024 values along a gamma curve that spans a higher range of contrast, but not all that much higher.