If you use a publicly trusted TLS certificate (either directly, or via Nabu Casa offering), domain will be recorded in Certificate Transparency logs.

You can slightly hide it if you use a wildcard domain in the TLS certificate (e.g. *.example.com), and then use a subdomain for the service. Ideally, something that's not home/hass/ha.example.com, or otherwise easily guessable.

Mini PC with N200 and NVMe SSD uses around 7W when idling.

For a minimally higher power consumption you can have up to 32 GB of memory, more powerful CPU, and decent GPU for video transcoding purposes.