Ok, the easiest way would be to simply cover up what you don't want to see, with an object the same color as your background. If bg is transparent, that won't work. Plus, I would strongly suggest not using this technique anyway.
Also, note there there will be a few different ways to approach this, and sometimes the choice comes down simply to personal preference.
The first thing that comes to mind for me, is to make 2 or 3 different, compound paths. However many different paths you need, and have them broken in different places, where you don't want the radial line to show. You can still use clones. The way I read your message, you'll need 3 of these paths, although I might have missed a detail here or there. Might need one or 2 more?
You could convert all the clones to paths, and edit them in place as them sit now. Or you could start a 2nd version beside this one (or maybe in another file, if necessary), create the specific compound paths, and recreate the radial array.
Or the idea which it seems you were working on, maybe making an opaque ring, or a ring with opaque spaces, to cover up the paths. That would work, only if the final version of this isn't going to have a transparent background. If you want it transparent, the only way to do it would be to edit the paths.
I can give you some tips on either way. I just hesitate to elaborate until I know which one to focus on.
And by the way, really nice work on that. I like it quite a lot! Oh and also, I'm not sure if you're aware, but it looks like you've been working at 35% zoom - meaning the the image is much larger than you realize.
It would take a lot of work, and I'm not sure if the outcome would be an improvement. But if each radial ray could come to a point in the middle, you might avoid that darkening effect in the middle. Oh! Or an easier way might be to give the stroke a gradient, so that the middle isn't as dark. Although I think it's interesting how you used that darkening effect and made the year characters white instead of black.
OH! Come to think of it, you could create a custom gradient for the strokes of the radial rays to accomplish your main goal. That would be easier than either of the suggestions I had!! And probably 3 or 4 gradients would be needed.
Anyway, let me know which way you want to go, and whatever specific info you might need.