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There could be a couple of things leaving you with problems. Firstly is that that the indexed shifter has a very slightly different pull ratio to that which is required by the mech which is why you're only experiencing some of the difficulties at the top and bottom end of the cassette. It's been a while since I've worked in a bike shop but cheap plastic thumbies were notorious for being awfully stiff at the top push.
Without seeing the cable outer length it's difficult to say for certain if it is long enough but if it makes one smooth curve from one end to the other and sits neatly in the boss/adjuster at each end then it's probably fine.
Does the penultimate ferrule sit in the braze on, or is the ferrule too big and so is sitting outside and on top of it so to speak? - This would have a significant impact on shifting accuracy.
I'm sure you've got it covered but the H/L screws should only just engage enough to prevent over-shift into the frame/spokes, if the L screw is set too far it will make the final push to get in to the largest sprocket even more difficult.
If you exhaust these possibilities or just don't care you could indeed swap to a 7 speed screw on block and get a matching 7 speed shifter and mech which would definitely all work together so any problem would be with cabling.
Oh one other thing that you could check just to be certain is that the mech hanger isn't bent.
Follow on from the last post about the right shifter / RD on my GFs bike.
After a bit of a ride the shifter keeps jumping back to the smaller sprockets. It can sit in the middle, but it won't stay in the largest two. Anyone got any idea why it won't hold?
Shit shifter?
Could the B tension screw have any affect?
Could the cable housing be a little short (I used the pre-cut length included in the pack)?
In terms of a solution, does anyone know if there is a better shifter I could use? Or could I upgrade to a 7 speed - my thinking is there are more shifter options? It has a screw on freewheel and 99% sure the rear spacing is 126mm (definitely not 130mm).