This kinda explains the problem, you can’t run a browser because it has no access to the system resources without a web server so your basically looking at a command line terminal

Considering that the host is running some *nix image, you would need to have a XServer running on the host system and then have a browser running somewhere using that XServer as the display.

HAOS does not provide an XServer and it does not seem possible to define a Docker container that would kind of “grap” the graphical card to manage its display output entirely – at least I did not find a solution to that. It is possible to give a Docker container access to GPU resources (search term: “docker gpu”).

So you would have to install a system providing an XServer Natively and then install Homa Assistent on top of that, either through virtualisation or containerisation.

You then likely want to have a browser run in Kiosk mode (hide the browser’s menus, not HA’s menus).

This forum has instruction on how to install HA using pip on a Raspberry Pi for Kiosk mode 66. Personnally I do not recommand installing HA Core (use PIP on the native system) – it’s harder to maintain.

If I were to try and do this, I would likely install Proxmox, and add an XServer to it 26. Then install HAOS in a container – there are even installation scripts for getting HAOS in a VM or container on proxmox 19 referenced in this installation guide on ow to setup HA on proxmox 27.
Once that is done, you need to setup the native system (where proxmox is running) to open a browser in kiosk mode (i.e. full screen, ensore that it is restarted every time it fails).