The good thing though is that those cheap $10 HDMI audio extractors work well for this use case if you have a playback device that outputs PCM over HDMI. As a side note those extractors are also a great way of getting 5.1 surround sound from a HTPC running the Dcaenc DTS encoder [1] into an old pre-HDMI AVR.
Because I'm pretty sure mine doesn't. I hate it, I'm the first to point it out or only one bothered by it. I haven't done anything (other than run Audyssey) to stop it, it just hasn't been a problem as far as I can tell.
And I know you're exaggerating, anyone would notice 'hundreds of milliseconds', but still.
Decoding, distributing, processing, and rendering across all the involved components can take in the order of 100's of ms. HDMI 1.3 and up has had a mechanism for equipment to communicate internal delays so audio remains time-aligned with the rendered image.
Some devices will also have manual overrides for this. If you are experiencing significant drift something is likely borked in the setup.
TV selection options go through the roof while analog loss and interference in the mixer is eliminated.