Just comparing logs, more or less line by line, I see the HE-AAC line in yours and the AAC line in mine. Does that mean you have set up the HE-AAC decoder but not the AAC decoder? That might make a difference if your audio track is AAC, although I can't imagine why it would play sound in recordings but not live. Greater minds?
I dunno, I disabled my AAC decoder and enabled my HE-AAC decoder (and indeed got no sound) but didn't get the HE-AAC line in my log. I don't know what one does to get that line, or if it makes any difference.
Not really NextPVR treats their playback differently and live tv is not very dynamic It is possible that the channel is configured for AC3 but I'd start by not using the generic system aac decoder.