Hello! To completely disable access to the browser, you have three main options, in order of effectiveness:
#1 Parental Controls
If you enable parental controls on your Switch, even if none of the content rating or categories are blocked, access to the Network Settings and editing WiFi hotspots will be locked behind a PIN. Without being able to access the settings, alternative DNS servers cannot be enabled, and the Connection Test browser can't be accessed.
How to Enable: https://www.nintendo.com/us/switch/parental-controls/
In my opinion, this is the intended method by Nintendo to prevent access to the built-in browser, among other network settings (like VPNs or proxies).
You can also explicitly disable the Social Media browsers (Facebook and Twitter) from the Parental Controls app.
#2 Router-Level Networking blocks
DNS requests can be overridden at any point in the networking connection chain. This would need an advanced networking setup that configured on the router. If these DNS packets are overridden, whatever DNS is entered on the device will be effectively ignored. This would only apply while using your network, though.
Essentially the opposite of this reddit thread: https://www.reddit.com/r/HomeNetworking/comments/q81j5v/is_there_a_method_to_force_dns_resolve_on_a/
You could also more simply only block the three IP addresses (45.55.142.122, 45.55.112.11, 46.101.65.164), if your router allows it.
#3 The "Disable Browser" (LaneChange) pages
There are three pages that control the different DNS addresses and what they will resolve for your IP:
http://lanechange-controller.fortheusers.org/manage/
http://lanechange-us.browsedns.net/manage/
http://lanechange-uk.browsedns.net/manage/
These take your current IP, and tell our systems to not perform the redirect on them. So this is more of an "opt-out". The reason this wears off over time usually has to do with your IP resetting or changing. And this is also only tied to the network, not the device.
Other Info
The Switch is a video game device first, but it can also play media (such as from Hulu, YouTube, or Twitch), and it utilizes different web browser applets across the system. Nintendo contracts with another company to maintain these browsers with security updates.
Switchbru and BrowseDNS allow access to one of those browsers (the Captive Portal applet) by redirecting connection test DNS requests. To get more specific, every time the Switch connects to the Internet, it performs a check to see if it's behind a Captive Portal, and if it is, it displays the browser. This is often necessary when authenticating your device with paid Wifi hotspots, like at a hotel or on an airplane.
There are other methods that don't involve our servers that will also allow this browser to display, such as using a self-hosted Captive Portal from another computer on the network.
For this reason, I think that the Parental Controls solution is the best approach for blocking the browser, in the same way you might want to prevent unrestricted access to apps like YouTube.
Last Things!
The intent of the DNS service is to help make the Internet more accessible, but I completely understand why there are scenarios where you would want to prevent that. I'm a big advocate for Nintendo fully exposing the browser's functionality and making it less hidden, and less "cloak and dagger". Any presence of a built-in browser should be clearly presented to the user.
An upsetting precedent is that the Playstation 5 browser works in a very similar manner. This is not the direction we should be heading in! If a console includes a browser, it should be up front about it, and allow the user to control access to it directly, alongside other apps and media. Like on the 3DS and Wii U!
I hope this fully answers the question, and provides enough information on both blocking the browser, and how the browser itself works.
