The recent Nintendo Switch Version 4.0.0 has been found to have added support for Wireless USB headsets. [Thanks, SoldierOfOrange on Reddit]
While this addition was not mentioned in the patch notes, the wireless USB headset support has been confirmed to include PlayStation wireless headsets as well as several Logitech models.
Meanwhile, Kotaku UK has reported that Nintendo wants to keep video capture limited to 30 seconds until every first party software supports the feature, and is also supported by larger third party games. Nintendo will also approach developers who have used their characters in third party games, such as Mario + Rabbids, in order to implement video capture earlier, although any developer can patch support in for their games.
After these conditions are met, Nintendo will add options for one, three, or five minutes through another firmware patch, which would work for all games with support, and accordingly will use more storage on the Switch. While there is no time frame on the implementation, future games may support video capture from the get-go.