If you provide the server, the actual computer used to provide a service, you are responsible for that server. Minecraft Bedrock servers are operated directly by Microsoft and as such are something they are responsible for.

The browser accesses any and all servers for web pages. Regulating the browser would not work as they have no control over the content, nor do they have the ability to effectively filter.

The confusion comes from you running two similar programs on your computer and having one of those behave differently to the other. The key is that the game client is a client only for that game, but the browser is a client for any web server out there. The client is not regulated at all, the server is. Anyone can access any web server from a browser, but only one server from a game client.

Interestingly WoW has custom servers which are not operated by Blizzard and as such Blizzard is not responsible for what happens on them. The operator of the server is the responsible party here and has to control their platform.