This app was mentioned in 10 comments, with an average of 1.50 upvotes
It costs a few bucks, but the app Kiosk browser is what I use.
It's got a ton of options, including hiding the status bar, the nav buttons if you have them, keeping the screen on, and a lot more.
It's well worth it and it's what is running our HASS setup sitting on my desk (soon to be replacing an unused 3gang light switch panel)
Depends on if you need to lock it down to an app or to a web page, i use https://play.google.com/store/apps/details?id=com.procoit.kioskbrowser right now. Just search for kiosk in the app store and start messing with them.
We've used this on several tablets and it worked well enough! I believe you purchase the full version license from their website and it was $20 per device.
https://play.google.com/store/apps/details?id=com.procoit.kioskbrowser&hl=en_US&gl=US
So, the first thing is: How do we create a kiosk mode?
With the Raspberry Pi it's obviously no problemo, so let's look at Android: As it turns out, there's also a way for Android-Apps.
Looking a bit further: There's actually an app for that!
So, assuming you have wifi: Download the app I linked, and just put up your webpage. Done!
Edit: With the website, I'm still not 100% clear what you want. Is the website already finished or not? If yes: What changes need to made, so it works fine with the requirements you have chosen? What's missing?
It's rather hard to tell you what to do, if we don't have an idea what you already have. Nobody's gonna look down on you if you post a link to what you're trying to replicate!
The Home Assistant Companion Apps use the themes installed in your HA system, since it's essentially using a web view to render the UI. Alternatively, something like KioskBrowser would also be using the web UI.
So you just need to look for an OLED-friendly theme in the community forums, or roll your own theme if you have the patience.
Damn, I had an app just for this that I believe uses webview (Chrome). Let me see if I can find it.
Edit: here
If you do choose kiosk mode on Android, this app is great, I have nothing but praise.
https://play.google.com/store/apps/details?id=com.procoit.kioskbrowser
might give this a try:
https://play.google.com/store/apps/details?id=com.procoit.kioskbrowser&hl=en
Here's one that seems pretty good: https://play.google.com/store/apps/details?id=com.procoit.kioskbrowser