Am I asking too much?
It seems to me that without that knowledge, you lose a lot of perspective on how software works. I think you also possibly miss opportunities to automate some of your most tedious tasks.
I write multiple testing scripts for our company's software tool that are immensely helpful and time-saving. No one else on our team has these kinds of scripting and automation skills and it can end up being frustrating. Many of the other GIS Analysts are surprised about what can be automated. I built a basic unit testing system for us that none of the programmers really had time to build. I'm basically going alone when I do these things.
I enjoy the scripting portion of my job more than the QA portion of my job by far. Perhaps the reason you don't find too many dedicated QA people who can write scripts is that, by and large, people who can script/code like doing that more than testing.
A QA analyst (someone who runs manual tests, fills out bug reports, and annoys programmers) does not need to be a programmer. It would be a waste of salary and it is more important that these people are persistent and meticulous than they know the internals of the system they are working with.