Thought I'd pop in and share some of my insights, too. I echo much of what's mentioned above by Dory and Elliott. I'm managing the Marketing Automation Operations for our US region.
I do have HTML experience and can get in and do custom coding - I can code responsive design, but I'm slower than most because I don't do it day-in and day-out like full-time developers do.
If you have the budget to rely on dedicated design resources, I think that's definitely the way to go. It's easy to get stuck in the day-to-day and lose sight of the bigger picture, so I believe the power users in an organization should really be there to help optimize, recommend best-in-class processes, integrate new tools (where applicable) to automate, data strategy, and run the operational/more complex programs. I think this is really where the value of a marketing automation/operations role comes in.
Certainly there will be trouble shooting and day-to-day things like basic HTML revision, etc. that you'll still need to manage. That said, if you're trying to learn responsive from scratch, you could use designers to do it faster and more efficiently while you focus on more strategic items.
Of course, you can still learn responsive for fun....I just wouldn't rely on it 100% of the time. Too much other work to do, hah! 🙂