I think part of your confusion is that you've mashed a load of separate issues together that don't really have any interaction with each other. The other part is that the way that pensions are paid out has changed.
The Civil Service Pensions are not funded, in that money paid out is paid by the treasury, rather than an invested fund. However, that doesn't mean that no one contributes to it. The 'employer' (remember that this could be several different organisations who are eligible for the scheme) pays a significant contribution (it's an average of 20% IIRC), and the member (now) pays between 4.6% and 8% of pensionable earnings (depending on your pay band). Based on your post I'm guessing that you left the civil service before member contributions were introduced*. However, the member contribution is nothing to do with the COPE.
The COPE situation arises because prior to 2016, pensioners received a state pension comprising 2 parts - the basic state pension and the state second pension. Having 35 years of NI contributions allowed you to receive full basic state pension. The second state pension was determined by a load of factors, including earnings and the specific NI rate that was paid. The term "contracting out" actually means that you have opted out of the second state pension by paying a lower rate of NI (and your employer also paying a lower rate of NI). This was only allowed if you were in a pension scheme of some sort, as the 'private' pension scheme would make up the shortfall.
To summarise, the way that your pension is funded or contributed to has nothing to do with the "contracting out". The fact that you have a 'private' pension has allowed you to 'opt out' of the additional state pension - or rather, it has allowed your employer to opt you out. The Contracted Out Pension Equivalent (COPE) value is the difference between the basic pension and the extra state pension; this is paid out as part of your 'private' pension. It's confusing because your pension statement won't list this as a separate amount.
This is all made more confusing because the way the state pension is paid changed in 2016. The new state pension is a combination of the basic and top up elements of the old pension. If you have full NIC years, the starting amount is £168 p/w. However, this starting amount is both the basic and top up of the old pension. Therefore, if you weren't eligible for that top up amount, it's taken off your starting amount. Under the old system, you wouldn't have gained the top up; under the new system you've lost the top up value from the estimated amount. It works out being the same thing, but it appears that something is being taken away from you.
TLDR - you aren't eligible for that money to be paid in state pension because you paid a lower rate of NI than people who have no other form of pension
* As a side note, Civil Service Pensions are also no longer final salary, and all active members were ported onto a defined benefit scheme in 2015, unless they were close to retirement.