Applicability of Presumptive taxation

Hope you had no business income in FY 2020-21.

In that case, AY 2022-23 is 5th year in continuation...

As said earlier, you are not required to continue it till next 5 years, but filing u/s. 44AD is mandatory for current year, otherwise tax audit will be applicable.

Is the tax under presumptive section for a partnership applied within the partnership and is tax free once transferred to the partners account?

The distribution of the profits of the firm after payment of taxes is tax free in the hands of partners.