I am probably over thinking this but....

Jane Smith as a personal individual account with the bank, John Doe is an authorized signer on the account. Sally Brown is a Power of Attorney for banking matters for John Doe.

Can Sally Brown sign checks and conduct banking business for Jane Smith because she has a POA for John Doe?

Thanks for your help.

No. Jane authorized John to sign on the account. She did not authorize him to let someone else sign on the account.

In general, fiduciary responsibilities cannot be delegated to another person.

Just to say it another way
John authorized Sally to do John's banking business - and it stops there, the POA does not carry on "another level".