I regularly test disconnecting my HA server from the network and make sure nothing home critical stops working.
I do have a few internet integrations (weather, upcoming sports events, etc) and those all stop when I do a “no internet” test, but Home Assistant runs just fine without any network access.
A fresh Home Assistant Operating System installation is useless without an Internet connection. It needs to download several gigabytes of Docker images for it to be usable.
It's very likely they'll take the infrastructure down if they sell their business or go out of business. Home Assistant OS will be completely useless in such a scenario.
The more likely scenario is that someone installs Home Assistant OS, sets it up with the current version and uses it as it is to avoid breaking changes. Their storage or the entire device breaks down. They'll want to install the same version to restore a backup. They'll learn the hard way that it's not possible to bring up what they had before.
Using the Docker image is the only choice which mitigates this risk.
The Home Assistant core container relies on resources hosted at https://brands.home-assistant.io. Those are cached in the browser for a while. Your mobile phone and web browsers still load the icons. They can do whatever they like with these bits of information. This includes selling allegedly anonymous Home Assistant usage data. The people who host their infrastructure can still do that without their knowledge since they claim it's a CDN.
The use of this CDN means that an existing Home Assistant core based setup can stop displaying some icons if those icons are removed. This can happen if someone decides to use a specific Home Assistant core version.
I ran the UI and didn't load anything from external source.
The Internet should only needed to download the image. You could download and flash it manually. The setup shouldn't require the Internet. What needed the Internet?
As I've written in the other comment, the Home Assistant Operating System is useless without an Internet connection. It has to pull the Docker images since it doesn't have any version of the images at all.
This thing isn't as private and as local as it is claimed to be. It's very tied to services provided by Nabu Casa, including the very visible cloud integration of theirs. They've expanded it now to backups.
I regularly test disconnecting my HA server from the network and make sure nothing home critical stops working.
I do have a few internet integrations (weather, upcoming sports events, etc) and those all stop when I do a “no internet” test, but Home Assistant runs just fine without any network access.