Tiger, I agree with all you post, except...
It may be a case that you have paid enough stamp to gain a full pension 
Highly unlikely. To get the full new state pension (of £155 ish) you need 35 years of NI contributions, AND to have not been contracted out of SERPS/S2P. Moggy has £26K in his pension pot from being opted out. The govt therefore make a deduction (called the COD - Contracted Out Deduction) from the £155 to account for the fact that his SERPS NI money was paid into a different (private) plan. This dramatically reduces the amount of the new state pension - often down to only £50-£60 per week.
There is a safety net - As at 6-April-2016 the govt calculated how much you would have got under the old state pension scheme. This was made up of £115 basic (if you had 30 years NI), plus any SERPS/S2P. Assuming Moggy has 30 years NI, but little or no SERPS/S2P (due to being opted out) then the old figure will almost always be higher than the new figure. It's highly unlikely that anyone with significant opted out years will be entitled to more than £115 + a bit of SERPS.
The starting amount is the higher of the old rules and new rules as at 5/6/2016. Going forward, you get £4.45 ish added to the starting amount for every post 2016 years NI you are credited with, up to the £155 maximum. So if you start at £115, and work/are credited with 8 years @ £4.45 you'll make the £155 maximum, AND get to keep the money/income in/from your private pension plan.
I know my dad stopped paying full NI well before he retired due to the amount he had paid in 
You can't stop paying NI on earnings unless/until you reach state pension age. It doesn't only pay for pensions - it helps pay for the NHS amongst other things. If you continue to work past state pension age, then its true that you don't pay NI.