I'm pretty sure you cant.

I double checked because I was pretty sure that you couldn't remove featured servers, and I was right. You can check on Google.

Other than that, you can add screen time to the iPhone/Samsung/Android/Laptop/iPad. If you don't have an option for screen time, then just limit their time on the devices they have Minecraft on.

For example, if you had a kid that loved playing Minecraft servers but there were public servers with cyber-bullies, just watch a video or read a newspaper and keep an eye on your child. This won't work 100%, but this is a tactic you can use.

You can also give them the device they play Minecraft on when they finish dinner or do chores or something that contributes.

-W

What you can do is change your network settings from DHCP to manual.

On a PC, you would -- start > run > ipconfig.

Record the IP and subnet.

Change the network settings to manual, and leave DNS servers empty. The device will no longer be able to hit the internet, but at least you'll have access to your LAN server.

Unless you're not running a LAN server - which you should.

So in this case, you could do some network detective work, and find the featured public server's IPs, and then block those. You could do these multiple ways, but you can google that for your specific device, or block the IPs on your firewall.

I Have a Solution!

I have a solution the might help! Try switching over to Minecraft Education Edition. Now this isn't very intuitive, or convenient but it works!

Here's Why

Minecraft Education Edition (or MCEE) does not show featured servers. You can move your worlds to MCEE if you're tired of the server list. This is because MCEE does not support servers outside of those made by people who published the IP, or code, to join their world (which even then, is only temporary). So if you are thinking of making the switch, here's how:

How To Make the Switch

1. Export the world you are currently on and put it onto your computer as a .mcworld file.

2. Make sure that the world is on, or is converted to 1.14.

3. Go to your computer files and open the .mcworld file, which will automatically open MCEE and begin importing the world into it.

4. Use creative mode to restore your inventory to its former glory and play from there.

Just To Be Safe

1. DONT make the world a .zip file or else nothing will work.

2. If you want a legitimate survival world make sure to go to the settings of the world before you open it, and disable cheats (and coding if you want).

Cons

1. The first downside is that if you wanted to play together at the same time you would need more computers

2. The second is that you need a school account or else it'll keep pressing you to try the MCEE exclusive coding lessons

3. The third is that MCEE is still on 1.14 (The bee update I believe).

Pros

You did say you wanted your children to somewhat learn and use their brains, right? Well here I have a list of things exclusive to MCEE.

1. Due to this version being named "Minecraft Education Edition" it has a variety of lessons on things like Math and science, but at the same time basic Minecraft things like redstone, building, brewing, and etc.

2. There is also the coding lessons exclusive to MCEE that i briefly included. They could, of course, teach your kids about coding, giving them a chance to make some fun personal mods that can do things like automatically build a big wall, or spawn a horde zombies with armor and swords for you to battle! But those are only a few of the possibilities you have with MCEE's coding system.

3. if you want your kids to unleash their creativity MCEE has different world templates that will allow your kids to do things like: Create the cover of a book, make a comic using NPC's and posters, build a spaceship, or just build a mars rover.

If Not...

Now if this seems a little extreme, or doesn't look like the right solution to you, that's fine with me just correct me or give me advice as to how this process could work better. Just be sure to not change too much if you edit this answer, just make changes like adding more tips or sections that might help, or correcting grammar and formatting mistakes.