[noggin]

And why the heck would it struggle with high-bitrate SDR content over internet but not over local network? (Speed tests on both ends indicate the network performance is way higher than necessary.)

Network latency and jitter can be significantly worse over WAN than LAN connections.  You may be able to mitigate this by increasing the buffer size that Kodi uses - possibly in the advanced settings xml file - but I've not used Android TV for this type of stuff so can't advise the route to do this.

I've seen other reports about Tonemapping causing issues with this platform.

I've retired this as a spare bedroom TV dongle to let visitors watch Netflix - had too many compromises as a main or secondary viewing platform for me. (Got a very heavy discount on it so I don't feel too bad. It's a pity - it's a nice piece of industrial design and the remote is OK)