It'd be wild if Switch just had a device that let you easily display the video feed somewhere else, like another screen of your choosing, some kind of docking system.
(I agree that forcing the icon to constantly display in the corner is dumb as hell, lmao)
Upon reflection, this was perhaps too much of a smartass reply, but to be fair, the OP was being rather hyperbolic.
Honestly, there's no 100% perfect solution for playing 4:3 aspect ratio on OLED screens. True black bars would allow the pixels on the borders to rest, but then you're wearing out the OLEDs in the middle of your screen faster. Gray bars without a gradient is probably the "best" solution, but even then, you might still be able to notice burn in in certain circumstances. Your best bet, other than the extreme option of playing on a 4:3 CRT monitor, is to play 4:3 games on a low latency LCD screen.