Call me an "incremental reformist" if you want, but I do think that the following would be a politically palatable start to reform the current SCOTUS selection.

1. Appeals Courts for each district nominate one member each among them for the president to pick from.
2. Constitutional Amendment to return the confirmation of all Federal judges to a 60 vote requirement (or a 60% equivalent of a larger Senate).
3. A one month mandatory FBI investigation into the candidates background and financial history to provide a preliminary account of any conflicts of interest.