Previously you download an app to the phone and can start control your devices, that is because there’s a server (or servers) in the cloud somewhere, which is subject to internet availability and the willingness of the device vendor(s) to keep running servers in the cloud in the foreseeably future.
With HA, that concept still applies. The difference is that with HA you can take the control back from the vendor(s), and run that servers locally inside your house, in a RPi or VM or repurposed laptop or some home server. There was a recent blog post about this.
So yes, the HA server would be always on.
Now to the “client” side, you can use a browser from any PC/laptop (see demo), you can use the HA app available on both iOS and Android, and if you want, you can even build your own community based dashboards… and not using the official one nor official app.
So back to your initial requirement - a touchscreen dashboard in the hallway… many people have done this also. Yes you can build a RPi with touchscreen and fullscreen browser to connect to your HA server, and you could also repurpose a tablet to do so… there is even “kiosk browser” to make things more configurable. I have also seen wild projects using a projector + voice control to replace touchscreen.
Hope this helps.