My Sony Bravia TV sound system is horrible, but my headset hyperX is sufficient.
I tried the apple LocalCast but it does not allow you to cast video at all from its browser.
Android: 10, latest.
Phone: OnePlus 6T.
Chromecast: 3 gen.
TV: some 10 year old Sony Bravia.
Robert's answer in comment:
When casting an online video your smartphone simply tells your TV
which web address to play. The audio is never processed by your
smartphone. Therefore it can not be played by the smartphone. Better
connect the headset to your TV.
So not possible the asked feature, but you need to connect your headset to your TV.
Q & A