You stop paying national insurance when you claim your state pension.
Age-related tax allowance, calculating tax if you carry on working, National Insurance payments
I think you'll find that you can stop paying NI when you reach State Pension age regardless of whether you claim the pension or not!.
Stopping paying is not automatic, you have to notify your employer.
So those that work beyond pension age can stop paying NI on their pay packet.
My wife could have got her pension at 60 but continued working until she was 63.
She also deferred her state pension because of the tax implications.
She claimed her state pension at 63 and was given a lump sum of three years state pension plus interest. (she could have had an enhanced weekly pension, but the lump sum was a better deal for her).
She paid no NI on her salary after her 60th birthday.
If you are thinking of working beyond state retirement age take professional advice.
She ended up a few thousand ££ better off.