I'd agree that the CIPD don't help by building an unwieldy framework around something that should actually be quite lean and nimble. You don't need a big HR team to make a positive difference.
I see the role as including:
Training and supporting new line managers, so they don't need too much day to day input from HR and can have confidence to lead teams
Coaching people to develop their careers and enjoy their work
Governance, H&S, legal compliance etc
Strategic planning for the skills we will need to recruit for in order to achieve company growth (some data analysis obvs)
Working on all sorts of things to keep the leaver rate low
Negotiating hard to keep recruitment costs down too
Setting up support systems aiming at low sickness rates because people can ask for help.
All of that falls under HR but in my view a lot of organisations just have way too many people doing it.
At director level you want to be in total command of the data in your own co, and your sector, so you can make valuable input into the co growth plans. You'll be lobbying for employee benefits that they will actually want. And you'll have a reputation as a problem solver, a reputable source of advice, and someone who will soak up the "people" side of some of the more difficult situations that inevitably arise at work. If you're doing all that then you've earned your place on the board, and your salary 
OP to address your recruitment question - always see if you can ring the hiring manager to talk informally before you apply. If they say yes, then prepare thoroughly for that call. Keep it brief, have a few focused questions, such as "this is a sector you're keen to break into, what are the key skills they really need in the role?" Show you've done a little research about the company and throw something in that you've learned. Thank them for their time, and then send in a polished application that makes a brief reference to the call in your cover letter. If you're still not getting anywhere, try calling the HR team and asking which agencies they like working with, then get yourself on that agency's books.