The web site you build does the same thing that the standalone app does. The web site is hosted by the web server. That's an added piece of software to maintain and configure, over and above the web site itself.

The web server is going to be fine. It's usually gone through a lot more testing than your display. Displays are trivially and commonly implemented as fullscreen webapps.

It's the browser javascript that I would worry about; responsiveness is always a battle.