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I think the easiest way would be to install a launcher that let's you restrict access to certain apps. First that comes to mind is Luna Launcher
It will only display apps that you want (unless you unlock it with a code) and you can lock the thing down to just do the tasks you want.
Google Family Link would be the most straightforward option. It lets you control what gets installed, block/allow apps, it lets the kids request apps that they would like installed for you to review, can set time limits, etc.
Another option might be Luna Launcher. It's a freemium app so you can try it before deciding to pay to unlock additional features.
I cannot emphasise enough the usefulness of Luna launcher in turning a generic tablet into a kid-friendly device. As for hardware, things to look for are:
You can expand your search by looking at used devices. This will put additional tablets (7-8in) as well as phablets (5.5-6in) into your price range.
First of all, go to the play store app, and access the play store settings from the burger menu. In the play store settings, disable automatic updates. Since Youtube is part of the Google umbrella, a set of default google applications will install themselves unless you specifically tell the system not to allow this.
Next, you'll want to find a launcher that is configurable to your wishes as a parent. Luna launcher is one such example, but there are plenty of options out there. Configure the launcher to display only the applications that you want your child to use.
Next, you'll want to figure out how to control their internet access. There are two ways to do this. The first is called "Whitelisting", which is a method you use to allow only a specific, known number of websites. The second is called "blacklisting", which works in the opposite fashion - blocking access only to those websites you specifically know about and wish to restrict. As a parent myself, I suggest the latter - too much enclosure and your children will undoubtedly find ways around your restrictions, not to mention that the opportunity for learning and growth is also restricted. However, an advantage to whitelisting may be restricting your child's contact with a specific person or group, which has its own benefits.
I personally look to enterprise solutions for managing internet access. Sometimes there is a learning curve, but usually there is plenty of documentation available to get up to speed. An example of a browser that supports blacklisting and keyword blocking, and is remotely administered through a parent or administrators machine, is Codeproof: https://play.google.com/store/apps/details?id=com.codeproof.androidbrowser
https://www.codeproof.com/blog/secure-internet-browser-and-website-content-blocking/
Lastly, remember that these are not foolproof methods. Children and teenagers are inventive, smart, cunning and unexpected. If you grew up with computers at school, consider what you might have done yourself to bypass web restrictions at the time. The methods here are an aid, not a final solution.
Here are a couple of launchers I found: Luna Launcher and Kid's Shell, I recommend trying to search for "Kid launchers" in the play store. They seem to be pretty good, but I'm not sure about tablet compatibility. This article showcases some apps your kid might like.
Hope this helps!