Maybe I have been missing your point, but I think you are missing a piece of the puzzle, most likely because I have not mentioned it previously. It appears to me that your examples have both pensions and salaries coming directly out of the same budget. I think there are government employers that do things that way, but the City of Seattle isn't one of them. In Seattle, the pension fund is separate from the general budget. Money in the pension fund can, I believe, only be used to pay benefits (i.e. pensions, disability benefits, payments to beneficiaries, and refund of contributions to non-vested employees who leave City employment) and the costs of operating the pension fund itself, i.e. paying the actuaries, the administrator and, I think, the retirement system staff (I'm not sure if these last are on the City's payroll or the Retirement System's). Theoretically speaking, the City could move money from the pension fund to the regular budget by deducting the employee contribution from employees' paychecks but "forgetting" to put the money into the pension fund, or by withholding some of the employer match, but this would be a violation of both the ordinance that established the retirement system and the City's contracts with employee unions. Assuming no lawbreaking or breach of contract on the part of the City, my replacement's salary would be paid out of the general budget, and that's why I didn't include it when estimating the effect of changes in my retirement date on costs to the pension fund. Have I got your point now?
In your two options, you only included my replacement's salary in option 1, so it's not an apples-to-apples comparison. Including the replacement's salary, the two options are:
Option 1: Retire at age 52 and draw pension immediately. Cost to pension fund=X. Cost to City budget for replacement's salary=2.35X (based on today's starting salary for someone with my job title, adjusted backwards by the COLA adjustments since the year I was 52). Overall cost=3.35X
Option 2: Work additional 8 years, then retire. Including the effect of COLA adjustments on base salary, even with low inflation, if I continued to work until age 60, my starting pension would not be ~1.5X, it would be a little more than 1.93X (using my actual salary). So, for option 2, Cost to pension fund=1.93X. Cost to City budget for replacement's salary=2.35X, plus COLA adjustments to starting salary. Using the same low guess at future CPI which I used to estimate my pension at age 60, the replacement's starting salary would go up by 9.5% between now and when I'm 60. That would make the starting salary for my replacement 2.85x, and the overall cost 4.78X.
But this comparison doesn't really tell us the total cost of providing the service, because I can't think of any way even to guess at the total cost of my replacement over time. There may be statistics on how long employees stay with the City, and on what proportion of employees stay in the same position for their entire career, vs get promoted, vs make a lateral transfer, but I have no idea where to obtain the data even if it exists, and that's what I'd need to depict a replacement with a typical employment history with the City.