I'm going to preface this by saying that this is how things worked for me as a new ref in Surrey and Essex, zero experience with your region.
I wouldn't worry too much about getting AR gigs early on to be honest - apart from running the line for my brother's team as a club AR a few times, I had 2 AR appointments in my first 2.5 seasons as a L9 & 7, and will have had another 5 or 6 by the time 6 months have passed after I became L6. I think that although many leagues are permitted to offer lines to L7's, it's pretty common that the comparatively small number of leagues who use NAR's will have enough L5's and L6's that they won't need to use L7's and below too frequently.
@Andy A, middles at grassroots level will be readily available to you even if you do just have to contact the leagues directly and offer your services. I think this is where you'll be expected to focus first. You'll be a better AR with experience in the middle and you'll be more "desirable" as an AR with the L6 qualification by your name if you do choose to work through the promotion scheme.