You shouldn't have to ream anything.
If the string fits, it fits.
It's the interface between ball end, the wrap and the block...
The best you could hope to do, is clean up the bottom of the counterbore leading to the main bore. Not sure a drill bit is going to get 'er done...
Reaming the main bore implies it was under-sized and/or you are accommodating looser ball end wraps. This can be a fine line, depending how loose the wrap is. You would be shooting for a diameter greater than the width of the loop, but less than the width of the ball end. If you waller out the hole too much your balls could get really stuck(twss).
Just wanted to throw some caution your way when you said "ream the bejeesus".
The problem, illustrated for those who don't savvy: