Chessbase 14 and Fritz 16 are different animals.
Chessbase 14 includes one of the Fritz engines and you can add other UCI engines, same as Fritz.
Read message #22 in this thread. Chessbase products can be confusing.
Chessbase 14 is foremost a database management program where as Fritz is more of a traditional GUI that includes a simplified database management tool.
Fritz has more analysis tools, with the exception of the more powerful database functions of Chessbase 14.
I use Chessbase 14 as well, I do prefer it over the Fritz interface for user analysis... I still like running the Fritz Full Analysis (engine analysis) on some games over the tactical analysis (engine analysis) on Chessbase 14.
If you want a program that you play games against chess engines Fritz is that program.
When you buy it, I assume you're getting both a UCI and an engine.
And as you said, yes, there are free UCIs too.
Personally I'd just get chessbase (yes it's a bit more expensive).
@jkh0208 mentions DGT boards, which I didn't think of (I've never messed with them).