I've been wanting this since the video capture feature was first announced. The system is already recording everything, but only keeps the last 30 seconds and doesn't save anything until the button is pressed.
So it shouldn't put any additional strain on the system, as long as your MicroSD is fast enough so it's not affecting the loading speed of the game (if the game is on the MicroSD) and possibly causing stuttering when the game is loading data.
It would require editing the system module that handles the recording, or writing a replacement for it, one where recording can be toggled on and off (and maybe change the current 30 second thing to a long press so that's still usable)
People have already been writing replacement modules for certain parts of the Switch OS, so I feel like this is bound to happen sooner or later.
Also, with the Switch's decent wifi speeds, streaming the live feed to a PC with low latency and good quality should be possible. NTR-CFW on 3DS did a very good job at that despite the 3DS having awful wifi speeds, but was just a bit too lacking to have decent FPS on both screens, and having the upper screen at 25+ FPS and the bottom screen at only 4 FPS looks rather bad, the Switch should be able to do it perfectly though.