When you gaze over this stupid topic....you have to wonder if the folks writing the Constitution (way over 200 years ago) could imagine the value put upon the Supreme Court today? My humble guess is NO.....they just didn't see the court as a third part of the Republic....otherwise, they would have wrote five to ten additional pieces into the Constitution.
As for adding three or five additional members? Once you do it, and the next new administration arrives....what would keep them from double-court-packing....adding three more members, or perhaps nine more members?
To be honest, with all the 'business' of the court....one might even talk over the subject and agree that nine different courts need to exist, and each have a different area of expertise.
An example....just have one single court for privacy of data or technology issues. Or having a court that only handles farm or agricultural issues? Or perhaps having one single court which handles the entertainment and music industry?
The need for the court to be based in the District? I don't see this necessity anymore. They could easily be moved out to Colorado or Oklahoma.
As for Biden getting the Democratic support to change the number....in the Senate? I don't see this happening. They certainly don't want a hornet's nest of three months of highly aggravating Senate work going on daily....trying to confirm three new judges. I would also add....if this were the gameplan....you need to start this by late summer 2021, and NOT in the spring of 2022 with primary elections underway in most states.
So I think it's mostly a doomed-to-die topic within the Senate, and likely to be one of the 300 promises of Joe Biden.....which never occur.