You can delete the internet browser from the built in apps, but if the reason behind this is to block or limit your kid from internet access, it will be pointless. Chances are, that your kid knows that you can re-flash firmware, basically wiping the System partition and re-acquiring it.
In comments, you did however mention that your kid does not have permission to install software on the Ubuntu laptop. This is not a restriction to flashing the ROM, because if one does not have root permissions, he can still install software locally, and that should be enough to flash something as far as I can tell.
A solution might be to make some sort of read-only partition, that not even root can edit, and on there, you would have a small program that counts, and limits through something like Xposed. I however doubt that the above would work and/or be practical, as you might need to make the program yourself, and I think that read-only partitions can be mounted as read-write.
Another possible solution would be to go to your router settings and limit the internet access there, however this is only a partial solution as you mentioned that you wanted to set up a counter that counts even when not at home.
If you have any questions regarding the router setup to achieve this goal, you can ask your question at Super User SE, where this would be more appropriate.