Can I base my 2021 Tax return on a 2020 amended return that has mailed but not processed?

You must.

No need to file an extension and wait for the amended return to be processed?

No.   It will be many months before your amended return is processed.   You do not have to wait for that in order to file 2021.

extension of time to file  is not an extension of time to pay the amount you owe.

If you expect a refund, after your 2020 1040-X is processed, there is no penalty for filing 2021 late.

It could be 2023 before you hear from the IRS about your amendment.