Does Chengdu social security agency handle it?

As long as you are the payer of social security, you don't need to pay the company. You can go to any one of the five local state-owned banks and any business outlet with your ID card. Social security cards can be used to see a doctor, see a doctor and buy medicines, and there are financial accounts in them that can be used as bank cards.

Further reading: How to buy insurance, which is good, and teach you how to avoid these "pits" of insurance.