As far as I know, this is some kind of recent (new-ish as of the Switch 2) check that blocks out the browser. I have had it appear before, but not often.
If you could start keeping a log of when it happens, that might be helpful to figuring out how it works. The DNS service was ran for a few years before even realizing the browser timeout was exactly 20 minutes, just due to a lack of research.
In theory, the "Feature is not available" notice is similar to what the console says when the browser is outdated, or hasn't been online regularly (as in, connects to Nintendo's servers) in a while. That's why I typically recommend trying to connect normally, using Google DNS, or updating.
If it's always happening around the same time, that might even point to a "quiet hours" kind of functionality, but I think that would be unlikely. The intended purpose of the Switch browser is to allow the user to connect to the Internet after all. If you have access to a PC, it'd be interesting to see if CaptiveDNS works during the timeout: https://github.com/browseDNS/CaptiveDNS