Replace broken tile with concrete
First that concrete subfloor you have is great.  Don't ever demo a good concrete bathroom subfloor unless you have to.  

Second if you don't want a tile floor it is very easy to demo the tile without hurting the concrete sub - How can I speed up the removal of stubborn tile?

Third - an easier more viable option is to take out an area of tile around tub including your broken tile and find complementary tile.  Since it is by the tub it is very conceivable that there could be accent tile there.  This would probably be $30-50 and a couple hours of work.