Should a database 'administrator' have the ability to deep dive into runtime query performance issues? I acknowledge that everyone's experience and abilities are different. Having said that, I want to avoid setting expectations too high (or too low) for a DBA that through actions; appears to be an 'administrator'.
Given:

I am a developer that is deep diving into SQL Server trouble shooting
and performance issues. At night I'm watching Brent Ozar videos with
a bucket of popcorn.
A company with several divisions, this division having around 100ish team members
Many customers with databases having millions of rows with ETL
A small DBA team that is stretched to handle these same customers. Handle HAG issues, backup, restores, create new deployments for new or upgrading customers.

I am NOT attempting to justify my own opinion. I wish to adjust either my opinion or others in the company.
Question:
Given the above, should expectations be that a DBA is simply an 'administrator'? Is this something you've typically seen?

I went into this soap opera (see my other recent questions) with the expectation that a DBA is 'expected' to deep dive. I have come to believe that I have misunderstood and I am leaning towards a DBA - being an 'administrator'.
I welcome others' experiences and perhaps advice on refining this question.
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.