but in the case of swing systems not necessarily
Back in the archives I talk a lot about IMA as a 'machine'. The interaction of the bits is important in the same way that the interaction of your car's camshaft with the valves is important.
Glad the action became more 'natural' with the correct grip and that the results were good. If you need more on the grip check out
Ask Scott #52.
One check on the lead hand grip - if it is correct then in the follow through position where the club is horizontal and your lead elbow is in front of your lead hip (and your body is in a proper impact position with your shoulders parallel to the target line), the club will not go past parallel to the target line (at least not without breaking your lead wrist). Also remember that the lead hand grip should be very firm - if you swing into high grass (like the fescue at the open) while holding the club with just your lead hand it should not twist in your grip.
Good luck
Peter