As far as I know, this isn't something on BrowseDNS's end, and it isn't yet understood why this error shows sometimes. It's good to hear that trying a different DNS fixed it though. Other things to try would be switching to Google DNS (8.8.8.8) and connecting normally, trying to update the switch, rebooting it, recreating the network settings, etc.
I don't have anything better to suggest at this time yet, unfortunately. Likely once someone in the community figures out specifically why this message is displayed, better advice can be given. It does appear that this "Feature is not available" message appearing like that was added in a recent OS update, after the Switch 2 launched.