Point 1:As a Technical Project Manager I've heard throughout my career that it is my technical knowledge that leads to my insights and problem identification skills. For years I questioned this belief, because I couldn't identify the specific skill I learned as a technologist that made me a good PM. Three years ago when I made a transition away from my technical core I discovered that it wasn't my technical skills that I was using. The skill I was using was interviewing. I asked questions of staff about what they had to do.
As a technical project manager it is assumed that you can identify issues and problems because of your technical skills. However, I believe that this ability to identify personel issues, resolve planning issues, and mitigate risks is actually more of an interviewing skill set. A good PM needs to be able to interview staff to determine how to proceed on any issue.
Trouble shooting and interviewing are not skills sets that all PMs have at the same level. If your just an ok interviewer you may need to rely on your own technical, process, or industry knowledge to fill in the gaps that you didn't get in your interviews.
A good PM is a person who understands their core skills and competencies and knows how to use them to find solutions to project issues. The destinction of which skills you use is less important than the ability to solve the problem. The competent PM can resolve the issues the incompetent can't.
Point 2:Most hiring managers won't be able to tell the difference in you during an interview so they'll be looking to see if you have solved the problems they have before or if you match the profile of someone that has.
Conclusion:The first question asked here was. Should I become more technical? Yes, do it by interviewing the staff of your projects and learning from them. If you get involved in your project's problems you'll become more technical.