As soon as Democrats get the trifecta, packing the court should be the first order of business. Adding ten seats to SCOTUS and confirming as left wing as possible justices in short order should be priority #1. After all, Republicans would do the exact same thing if the roles were reversed.
Court packing isn't really credible as it would just lead to nullification from red states.
Not to mention that court packing would necessarily be answered tit for tat, reducing the judiciary to a strict subordinate of the executive/legislative. It is a game that has no positive outcome...or really any conceivable outcome other than discrediting the independence of the judiciary and gutting the checks and balances that exist on the power of government.
As for your own proposal, I like it in principle, but the problem with the 2/3rds requirement is that I don't see party loyalty giving way any time soon; applied in itself, it would just lead to long-term gridlock where seats never get filled. There would need to be some mechanism that would fill the seats automatically if too much time passed, a la the Missouri Plan. In that situation, the politicians would be encouraged to work together to maintain some semblance of control over the situation.