Chrome could have 100% market share and it wouldn't necessarily be a problem. If I understand it correctly, Google would have to abuse their market position in some way.
Various other sites can get added to the whitelist if their videos are big enough and the user plays them often enough. But YouTube starts out on the whitelist. It's possible that dailymotion does too. The BBC, doesn't seem to be on the whitelist, though.
I really doubt this is the reason; for starters, the update of windows 10 i have now removed the option to disable Bing in the start menu, nor can I change the search provider. They're headed right back to where they used to be.
If they are going Chromium based, its only to ensure everyone uses their chromium browser