Does a DBA need to know how to program in a system language besides SQL?
It depends.

In a big shop, maybe not because you have 1000s of servers to look after and your tools are provided. In a small shop, you'll probably need more knowledge because you have a wider remit.

Scripting languages (PowerShell, cmd.exe for example) are always useful though for monitoring, deploying etc. I've often had some Perl scripts I had to maintain (just about). Then there are the assorted ETL packages that you'd be expected to know about.

Saying that, most DBAs (that I know) would be able to write basic CLR stuff or know PL/SQL well. For me, the dividing line is knowing about the broader patterns or architecture of .net or Java. I don't need it as a DB developer or DBA. In the same way that .net or PHP or Java Monkeys don't understand database design or architecture or code as well as I do.

Personally, I decided to stop chasing the latest or best client language years ago and focus on database work. That doesn't make me a "non-programmer": I'd need to learn it again if I had to.