Should a database 'administrator' have the ability to deep dive into runtime query performance issues?
Titles are just that, nothing more than a set of words, and the relevancy of their meaning are dependent on how you define that meaning within the context of your domain. In this case that domain is your company, so you guys should define the DBA role to fit the needs of the business (realistically under the umbrella of what a DBA can do).
DBA is one of those kinds of roles that I've found to evolve as a many hats or crossover type of position. The reasoning being is it's a career heavily populated by a mix of people who either were Software Developers, System Administrators, or neither, in a past life.
As someone who went from Software Developer to resident DBA to actual DBA at another company and back to a quasi-Software Developer / DBA now, I got to experience most of the database development and performance tuning side of being a DBA as opposed to server and database management - though I have and still do manage some of those aspects as well. This was because of my prior experience as a developer and because the companies I worked for needed someone who could write code but also help optimize performance in the database layer both from an architectural standpoint and a query tuning one.
So in summary, there is a rainbow of DBA roles out there between ones who essentially just administrate database systems and get their hands dirty under the hood of the servers themselves, others who purely do database design and performance tuning, and others who have a mix of actual software development (even outside the database layer) responsibilities and do performance tuning and database management.