When using BrowseDNS or Switchbru DNS to access the browser, video playback is unavailable with the following message: "Support Code: 2809-1212 This feature is not available".
However, videos do work elsewhere on the system (eg. the eShop, News app, or even YouTube app). The Switch has several different browser web applets on it, each with different permissions. Here's a table from the Switchbrew Wiki:

Custom DNS settings can allow the last one, WifiWebAuthApplet, to allow the user to browse the Internet. This applet has no whitelist, and allows any URL to be visited. However, it does not allow video playback. This is done intentionally by Nintendo to limit the usefulness of playing videos when using a custom DNS. (For more on how BrowseDNS works, see this this post, or this one for self hosting).
Unfortunately, that means video playback only works if you can serve it from an Applet that only allows a short list of whitelisted domains. As all of these domains are https, there is no chance to bypass this using a custom DNS server.
As of v13.2.0, the whitelisted domains include: nintendo.com, twitter.com, facebook.com and google.com. You would have to have control over a domain or subdomain from one of these to allow video content to play. Previously, sites.google.com could be used (aka SwitchTube), however this is now explicitly blacklisted as well.
The only real hope to get this functionality working is to publicly let Nintendo know that there is user demand for a standalone web browser (like on Wii, Wii U, DS, and 3DS) that supports these features.
If you have a homebrewed Switch, the app BrowseNX also allows browser video playback, but that is becoming increasingly rarer and unfortunately does not match the ease of use of an officially available browser.