Ethical hacking vs. security testing
First, let's take some "definitions" to start with.
From EC Council (emphasis mine)
A Certified Ethical Hacker is a skilled professional who understands and knows how to look for weaknesses and vulnerabilities in target systems and uses the same knowledge and tools as a malicious hacker, but in a lawful and legitimate manner to assess the security posture of a target system(s).
Security testing is a testing technique to determine if an information system protects data and maintains functionality as intended.
From the emphasis you can already spot a difference: ethical hacking is looking for weaknesses, whereas in Security Testing you're looking whether the defined security measures have been applied.
In Ethical Hacking, you're using the information available from outside (black box), whereas in Security Testing you can also use information available inside your company (white box).
IMHO, Security Testing is broader because hacking can be a part of the test. You can e.g. also check the Backup procedure, which is important for the availability after a data loss, but hackers are usually not interested in that topic.
One task vs. many tasks
It depends on how much information fits into your brain. Some people can remember lots of things, other can't. Some of my colleagues, manual testers in QA, are very good in what they do, but are easily overloaded with extra tasks, e.g. when it comes to programming / test automation.
On the other hand side, I do manual testing, programming, debugging, test automation, I deal with virtualization, maintain Bugzilla, have set up our Wiki, experiment with Raspberry and I'm now attending online classes for Kali Linux and Metasploit. And I still have fun doing everything.
Expert vs. generalist
When you concentrate on one topic, it's likely that you become an expert in that topic. This is what some people like to be. If you want to become an expert, I recommend doing that for a technology rather than a tool. Tools become obsolete and a new version of a tool may require you to re-learn everything. Think of the change from Office 2000 to 2007. That hit many experts hard. It also means you can only do one job, so it'll be harder for you to leave the company.
When you know many things, you become a generalist. You can combine things and see the associations between them. Just make sure you know enough about those many topics. Do not just spend one month to learn it, spend at least 8 months on a topic so that you have enough knowledge to find out whether someone tells you a lie or not.
The Pluralsight course Learning technology in the information age by Dan Appleman was really helpful in deciding what I want to learn. If you don't have an account, you might use the trial period to watch it.