The Nintendo Switch System Software update 4.0 (released 2017-10-18) introduced the ability to record the previous 30 seconds of gameplay:
Added the following system functionality:
- Capture video on select games
- To capture video, hold down the Capture Button during gameplay
- Up to a maximum of the previous 30 seconds will be saved in the Album. You can trim the beginning and end of each clip, and post to Facebook and Twitter.
- As of October 18, 2017, this feature is compatible with The Legend of Zelda: Breath of the Wild, Mario Kart 8 Deluxe, ARMS, and Splatoon 2
So there's likely no hardware-limitation to record gameplay. Nintendo probably limits recordings to 30 seconds to avoid filling the (very limited) storage, and requires each game to support the feature individually to free the necessary RAM for the last 30 seconds of video.
Streaming, meanwhile, requires high-bandwidth communication. Unless the console has a dedicated chip for networking or reserves a few of its CPU cores for the system, open connections produce an overhead which affect CPU speeds. This is why many low-end devices have difficulties reaching high speeds when under heavy load. To ensure downloads won't slow down games, the Switch pauses downloads when a game is running.
So while there's no hardware limitation, there's a design limitation. The Switch is, in theory, fully capable of streaming its gameplay. It just won't, because streaming would negatively affect gaming performance.