The reason it does that is so that you don't think there is a network issue or a fault and try and report it as such to BT so in answer to your question... no.
If you use a different router, than you will get a page not found message, which you may prefer.
For example the TP-Link TD-W9970 blocks Internet access using a firewall which can be programmed to a schedule in 30 minute time blocks.
You can make this blocking selective to certain websites if you wish. Its 100% configurable.
You can block certain game sites if needed.
Communityhelp (Private website) Lots of additional information and help.
You will need to be connected to your home BT Broadband connection to view it, or any links to it that I post on this forum.