It doesn't work like that.
Many job skills are generic, especially in administration, IT, finance and HR areas, to all sorts of industries - and the public sector is but one industry sector.
If you are technically good at what you do and have demonstrable interpersonal skills, there are many industry sectors where administration skills (your example) are transferable.
Having recruited for the private sector for many years, one comes to realise that you can usually spot the 'probably not worth interviewing' candidates from what is written on the CV. If a public sector employee has been doing the same thing for 20 years at the same sort of level that suggests something about them to me - but the same is true of someone applying as a private sector candidate.
If one can demonstrate organisational progression within the job whilst the organisation has been subject to change (not necessarily via promotion), then that's a good thing. And many public sector organisations have been subject to much change over last 10 years - with many individuals working in it thriving through it. These are the people who do well in the private sector.