In more detail -
One of the problems was probably that there was no requirement for reliability or maintainability in the original spec.
Vibrations
For many years, Westlands would not let anyone do any mass balancing of the main rotor blades. The smoothest Lynx I flew had one blade that was one ounce different than the other three, whil I saw up to 8 oz of weight difference on opposite blades on a very rough machine - the blades weigh about 95 lb if I remember right, so 0.5% difference in weight is substantial. That was in the mid-80's, so I don't know about now.
AFCS
The story I heard was the money for the AFCS was mostly spent in development, and little was left over for buying the actual hardware, so missile gyros (renowned for long life) were used. I don't remember who did the AFCS for the military version, but Louis Newmarks (now Smith Newmarks) did the AFCS for the civil Westland 30, which did it's entire development and certification flight test without an AFCS problem. Does that tell you something about reliability of the AFCS?
Engines
Why the Lynx didn't have particle separation built in as standard (even on the Naval variant) was a substantial oversight on the part of MOD. The A129 has a particle separator and must be one of the reasons why the same engine works so well in that helicopter, but not in the Lynx.
And those are just the things I can remember 25 years later!