This app was mentioned in 4 comments, with an average of 1.50 upvotes
You can use KioWare - like some of the others mentioned below, it is kiosk software to secure your devices to a single application (or multiple, if you so choose). It can be deployed by XML, and it does allow you to keep it on.
You can download a free demo and then license if you decide you are ready to purchase.
You're spot on with kiosk software. You can set to a single application (or more than one). Most options out there currently are perpetual (so not monthly) and per device. Quantity discounts apply for all, and you can see that there are annual support options as well.
This is a purposed device deployment and should be really simple to set up and deploy.
Most have free demos.
Some options:
KioWare Lite for Android : Demo download here: https://play.google.com/store/apps/details?id=com.adsi.kioware.client.mobile.app
Surelock/42gears : https://play.google.com/store/search?q=kiosk+software
Sitekiosk/Provisio : https://play.google.com/store/search?q=kiosk+software
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Good luck!
Most MDM solutions will be able to handle this for you. Here are some you could consider if you went that route:
The ones above I’ve touched and know work. I’ve never implemented an app to lock down the device but I’ve seen/heard of these: