I found the easiest solution around it (and probably the most efficient one, given that switching to other browsers or visiting websites that "purify" Youtube videos doesn't work for me). Furthermore, this solution does not need any additional apps, but simply... signing out of Youtube.
The solution is as simple as this:
-Access Youtube on Safari.
- Switch to Desktop mode*
*(click on 3 little dots on top-right side of screen-> option "Desktop").
-SIGN OUT** of Youtube
**(top right corner, little round icon -> option "Sign out").
- Reload Youtube, you are now a guest.
- Search for video you wish to watch. Press play, then pause, go inside another app via double click.
- Bring up Control Panel, click on Play***
***(swish to right page of Control Panel to find Video/Music Player)
And BOOM, it works!!! Youtube plays on background, Youtube plays if one locks the screen.
Hope this helps.
I assume it may also work with Youtube App, but couldn't be bothered to check. It should though, as it seems that background play is limited by YouTube for people who made the "mistake" of signing in...
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Tested on iOS version 10.3.1