It would be unusual for a TV to have a digital audio input. Most just have a digital output which is tied to the internal tuner. If there's sound on the HDMI, then the set will get its audio there. Many sets have a stereo pair associated with an HDMI input which allows for audio when connected to a component with a DVI output.
In general, TV sets are not designed for the kind of switching you see in a receiver. Each input has video and audio and you can't mix and match.
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